updating week 46
This commit is contained in:
mhjensen
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
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<meta name="description" content="Week 46: Support Vector Machines">
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<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
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<title>Week 46: Support Vector Machines</title>
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<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
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<!-- Bootstrap style: bootstrap -->
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Support Vector Machines, overarching aims', 2, None, '___sec0'),
|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
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||||
('A $p$-dimensional space of features', 2, None, '___sec3'),
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('The two-dimensional case', 2, None, '___sec4'),
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||||
('Getting into the details', 2, None, '___sec5'),
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||||
('First attempt at a minimization approach', 2, None, '___sec6'),
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||||
('Solving the equations', 2, None, '___sec7'),
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||||
('Code Example', 2, None, '___sec8'),
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||||
('Problems with the Simpler Approach', 2, None, '___sec9'),
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||||
('A better approach', 2, None, '___sec10'),
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||||
'sections': [('Overview of week 46', 2, None, '___sec0'),
|
||||
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
|
||||
('Hyperplanes and all that', 2, None, '___sec2'),
|
||||
('What is a hyperplane?', 2, None, '___sec3'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec4'),
|
||||
('The two-dimensional case', 2, None, '___sec5'),
|
||||
('Getting into the details', 2, None, '___sec6'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec7'),
|
||||
('Solving the equations', 2, None, '___sec8'),
|
||||
('Code Example', 2, None, '___sec9'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec10'),
|
||||
('A better approach', 2, None, '___sec11'),
|
||||
('A quick Reminder on Lagrangian Multipliers',
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2,
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None,
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'___sec11'),
|
||||
('Adding the Multiplier', 2, None, '___sec12'),
|
||||
('Setting up the Problem', 2, None, '___sec13'),
|
||||
('The problem to solve', 2, None, '___sec14'),
|
||||
('The last steps', 2, None, '___sec15'),
|
||||
('A soft classifier', 2, None, '___sec16'),
|
||||
('Soft optmization problem', 2, None, '___sec17'),
|
||||
('Kernels and non-linearity', 2, None, '___sec18'),
|
||||
('The equations', 2, None, '___sec19'),
|
||||
('The problem to solve', 2, None, '___sec20'),
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||||
("Different kernels and Mercer's theorem", 2, None, '___sec21'),
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||||
('The moons example', 2, None, '___sec22'),
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||||
'___sec12'),
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||||
('Adding the Multiplier', 2, None, '___sec13'),
|
||||
('Setting up the Problem', 2, None, '___sec14'),
|
||||
('The problem to solve', 2, None, '___sec15'),
|
||||
('The last steps', 2, None, '___sec16'),
|
||||
('A soft classifier', 2, None, '___sec17'),
|
||||
('Soft optmization problem', 2, None, '___sec18'),
|
||||
('Kernels and non-linearity', 2, None, '___sec19'),
|
||||
('The equations', 2, None, '___sec20'),
|
||||
('The problem to solve', 2, None, '___sec21'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
|
||||
('The moons example', 2, None, '___sec23'),
|
||||
('Mathematical optimization of convex functions',
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||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('How do we solve these problems?', 2, None, '___sec24'),
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('A simple example', 2, None, '___sec25'),
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||||
('Back to the more realistic cases', 2, None, '___sec26')]}
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'___sec24'),
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||||
('How do we solve these problems?', 2, None, '___sec25'),
|
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('A simple example', 2, None, '___sec26'),
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('Back to the more realistic cases', 2, None, '___sec27')]}
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end of tocinfo -->
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<body>
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@@ -103,7 +104,7 @@ MathJax.Hub.Config({
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<span class="icon-bar"></span>
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<span class="icon-bar"></span>
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</button>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
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</div>
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<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
||||
<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
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||||
@@ -156,7 +158,7 @@ MathJax.Hub.Config({
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<div class="jumbotron">
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<center><h1>Week 46: Support Vector Machines</h1></center> <!-- document title -->
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<center><h1>Week 46: Gradient Boosting Summary and Support Vector Machines</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Morten Hjorth-Jensen -->
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@@ -172,7 +174,7 @@ MathJax.Hub.Config({
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p>
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<center><h4>Sep 16, 2020</h4></center> <!-- date -->
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<center><h4>Nov 8, 2020</h4></center> <!-- date -->
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<br>
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<p>
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@@ -196,7 +198,7 @@ MathJax.Hub.Config({
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<li><a href="._week46-bs008.html">9</a></li>
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<li><a href="._week46-bs009.html">10</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week46-bs027.html">28</a></li>
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<li><a href="._week46-bs028.html">29</a></li>
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<li><a href="._week46-bs001.html">»</a></li>
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</ul>
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||||
<!-- ------------------- end of main content --------------- -->
|
||||
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@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Support Vector Machines, overarching aims', 2, None, '___sec0'),
|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec3'),
|
||||
('The two-dimensional case', 2, None, '___sec4'),
|
||||
('Getting into the details', 2, None, '___sec5'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec6'),
|
||||
('Solving the equations', 2, None, '___sec7'),
|
||||
('Code Example', 2, None, '___sec8'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec9'),
|
||||
('A better approach', 2, None, '___sec10'),
|
||||
'sections': [('Overview of week 46', 2, None, '___sec0'),
|
||||
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
|
||||
('Hyperplanes and all that', 2, None, '___sec2'),
|
||||
('What is a hyperplane?', 2, None, '___sec3'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec4'),
|
||||
('The two-dimensional case', 2, None, '___sec5'),
|
||||
('Getting into the details', 2, None, '___sec6'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec7'),
|
||||
('Solving the equations', 2, None, '___sec8'),
|
||||
('Code Example', 2, None, '___sec9'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec10'),
|
||||
('A better approach', 2, None, '___sec11'),
|
||||
('A quick Reminder on Lagrangian Multipliers',
|
||||
2,
|
||||
None,
|
||||
'___sec11'),
|
||||
('Adding the Multiplier', 2, None, '___sec12'),
|
||||
('Setting up the Problem', 2, None, '___sec13'),
|
||||
('The problem to solve', 2, None, '___sec14'),
|
||||
('The last steps', 2, None, '___sec15'),
|
||||
('A soft classifier', 2, None, '___sec16'),
|
||||
('Soft optmization problem', 2, None, '___sec17'),
|
||||
('Kernels and non-linearity', 2, None, '___sec18'),
|
||||
('The equations', 2, None, '___sec19'),
|
||||
('The problem to solve', 2, None, '___sec20'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec21'),
|
||||
('The moons example', 2, None, '___sec22'),
|
||||
'___sec12'),
|
||||
('Adding the Multiplier', 2, None, '___sec13'),
|
||||
('Setting up the Problem', 2, None, '___sec14'),
|
||||
('The problem to solve', 2, None, '___sec15'),
|
||||
('The last steps', 2, None, '___sec16'),
|
||||
('A soft classifier', 2, None, '___sec17'),
|
||||
('Soft optmization problem', 2, None, '___sec18'),
|
||||
('Kernels and non-linearity', 2, None, '___sec19'),
|
||||
('The equations', 2, None, '___sec20'),
|
||||
('The problem to solve', 2, None, '___sec21'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
|
||||
('The moons example', 2, None, '___sec23'),
|
||||
('Mathematical optimization of convex functions',
|
||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('How do we solve these problems?', 2, None, '___sec24'),
|
||||
('A simple example', 2, None, '___sec25'),
|
||||
('Back to the more realistic cases', 2, None, '___sec26')]}
|
||||
'___sec24'),
|
||||
('How do we solve these problems?', 2, None, '___sec25'),
|
||||
('A simple example', 2, None, '___sec26'),
|
||||
('Back to the more realistic cases', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
||||
<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
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</ul>
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<a name="part0001"></a>
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<!-- !split -->
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<h2 id="___sec0" class="anchor">Support Vector Machines, overarching aims </h2>
|
||||
<h2 id="___sec0" class="anchor">Overview of week 46 </h2>
|
||||
|
||||
<ul>
|
||||
<li> <b>Thursday</b>: Summary of Gradient Boosting and further examples of applications.</li>
|
||||
<li> <b>Friday</b>: Support Vector Machines, classification and regression</li>
|
||||
</ul>
|
||||
|
||||
Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) of Hastie et al contains also a good discussion.
|
||||
|
||||
<p>
|
||||
A Support Vector Machine (SVM) is a very powerful and versatile
|
||||
Machine Learning method, capable of performing linear or nonlinear
|
||||
classification, regression, and even outlier detection. It is one of
|
||||
the most popular models in Machine Learning, and anyone interested in
|
||||
Machine Learning should have it in their toolbox. SVMs are
|
||||
particularly well suited for classification of complex but small-sized or
|
||||
medium-sized datasets.
|
||||
|
||||
<p>
|
||||
The case with two well-separated classes only can be understood in an
|
||||
intuitive way in terms of lines in a two-dimensional space separating
|
||||
the two classes (see figure below).
|
||||
|
||||
<p>
|
||||
The basic mathematics behind the SVM is however less familiar to most of us.
|
||||
It relies on the definition of hyperplanes and the
|
||||
definition of a <b>margin</b> which separates classes (in case of
|
||||
classification problems) of variables. It is also used for regression
|
||||
problems.
|
||||
|
||||
<p>
|
||||
With SVMs we distinguish between hard margin and soft margins. The
|
||||
latter introduces a so-called softening parameter to be discussed
|
||||
below. We distinguish also between linear and non-linear
|
||||
approaches. The latter are the most frequent ones since it is rather
|
||||
unlikely that we can separate classes easily by say straight lines.
|
||||
<a href="https://www.youtube.com/watch?v=efR1C6CvhmE&ab_channel=StatQuestwithJoshStarmer" target="_self">Overview of Support Vector Machines</a>. see also <a href="https://www.youtube.com/watch?v=N1vOgolbjSc&ab_channel=AliceZhao" target="_self">this video</a>.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -200,7 +184,7 @@ unlikely that we can separate classes easily by say straight lines.
|
||||
<li><a href="._week46-bs009.html">10</a></li>
|
||||
<li><a href="._week46-bs010.html">11</a></li>
|
||||
<li><a href="">...</a></li>
|
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<li><a href="._week46-bs027.html">28</a></li>
|
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<li><a href="._week46-bs028.html">29</a></li>
|
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<li><a href="._week46-bs002.html">»</a></li>
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</ul>
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<meta name="description" content="Week 46: Support Vector Machines">
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<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
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<title>Week 46: Support Vector Machines</title>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,90 +155,36 @@ MathJax.Hub.Config({
|
||||
<a name="part0002"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec1" class="anchor">Hyperplanes and all that </h2>
|
||||
<h2 id="___sec1" class="anchor">Support Vector Machines, overarching aims </h2>
|
||||
|
||||
<p>
|
||||
The theory behind support vector machines (SVM hereafter) is based on
|
||||
the mathematical description of so-called hyperplanes. Let us start
|
||||
with a two-dimensional case. This will also allow us to introduce our
|
||||
first SVM examples. These will be tailored to the case of two specific
|
||||
classes, as displayed in the figure here based on the usage of the petal data.
|
||||
A Support Vector Machine (SVM) is a very powerful and versatile
|
||||
Machine Learning method, capable of performing linear or nonlinear
|
||||
classification, regression, and even outlier detection. It is one of
|
||||
the most popular models in Machine Learning, and anyone interested in
|
||||
Machine Learning should have it in their toolbox. SVMs are
|
||||
particularly well suited for classification of complex but small-sized or
|
||||
medium-sized datasets.
|
||||
|
||||
<p>
|
||||
We assume here that our data set can be well separated into two
|
||||
domains, where a straight line does the job in the separating the two
|
||||
classes. Here the two classes are represented by either squares or
|
||||
circles.
|
||||
The case with two well-separated classes only can be understood in an
|
||||
intuitive way in terms of lines in a two-dimensional space separating
|
||||
the two classes (see figure below).
|
||||
|
||||
<p>
|
||||
The basic mathematics behind the SVM is however less familiar to most of us.
|
||||
It relies on the definition of hyperplanes and the
|
||||
definition of a <b>margin</b> which separates classes (in case of
|
||||
classification problems) of variables. It is also used for regression
|
||||
problems.
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC, LinearSVC
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> SGDClassifier
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'axes.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">14</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'xtick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'ytick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
<p>
|
||||
With SVMs we distinguish between hard margin and soft margins. The
|
||||
latter introduces a so-called softening parameter to be discussed
|
||||
below. We distinguish also between linear and non-linear
|
||||
approaches. The latter are the most frequent ones since it is rather
|
||||
unlikely that we can separate classes easily by say straight lines.
|
||||
|
||||
|
||||
iris <span style="color: #666666">=</span> datasets<span style="color: #666666">.</span>load_iris()
|
||||
X <span style="color: #666666">=</span> iris[<span style="color: #BA2121">"data"</span>][:, (<span style="color: #666666">2</span>, <span style="color: #666666">3</span>)] <span style="color: #408080; font-style: italic"># petal length, petal width</span>
|
||||
y <span style="color: #666666">=</span> iris[<span style="color: #BA2121">"target"</span>]
|
||||
|
||||
setosa_or_versicolor <span style="color: #666666">=</span> (y <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">|</span> (y <span style="color: #666666">==</span> <span style="color: #666666">1</span>)
|
||||
X <span style="color: #666666">=</span> X[setosa_or_versicolor]
|
||||
y <span style="color: #666666">=</span> y[setosa_or_versicolor]
|
||||
|
||||
|
||||
|
||||
C <span style="color: #666666">=</span> <span style="color: #666666">5</span>
|
||||
alpha <span style="color: #666666">=</span> <span style="color: #666666">1</span> <span style="color: #666666">/</span> (C <span style="color: #666666">*</span> <span style="color: #008000">len</span>(X))
|
||||
|
||||
lin_clf <span style="color: #666666">=</span> LinearSVC(loss<span style="color: #666666">=</span><span style="color: #BA2121">"hinge"</span>, C<span style="color: #666666">=</span>C, random_state<span style="color: #666666">=42</span>)
|
||||
svm_clf <span style="color: #666666">=</span> SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"linear"</span>, C<span style="color: #666666">=</span>C)
|
||||
sgd_clf <span style="color: #666666">=</span> SGDClassifier(loss<span style="color: #666666">=</span><span style="color: #BA2121">"hinge"</span>, learning_rate<span style="color: #666666">=</span><span style="color: #BA2121">"constant"</span>, eta0<span style="color: #666666">=0.001</span>, alpha<span style="color: #666666">=</span>alpha,
|
||||
max_iter<span style="color: #666666">=100000</span>, random_state<span style="color: #666666">=42</span>)
|
||||
|
||||
scaler <span style="color: #666666">=</span> StandardScaler()
|
||||
X_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>fit_transform(X)
|
||||
|
||||
lin_clf<span style="color: #666666">.</span>fit(X_scaled, y)
|
||||
svm_clf<span style="color: #666666">.</span>fit(X_scaled, y)
|
||||
sgd_clf<span style="color: #666666">.</span>fit(X_scaled, y)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"LinearSVC: "</span>, lin_clf<span style="color: #666666">.</span>intercept_, lin_clf<span style="color: #666666">.</span>coef_)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"SVC: "</span>, svm_clf<span style="color: #666666">.</span>intercept_, svm_clf<span style="color: #666666">.</span>coef_)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"SGDClassifier(alpha={:.5f}):"</span><span style="color: #666666">.</span>format(sgd_clf<span style="color: #666666">.</span>alpha), sgd_clf<span style="color: #666666">.</span>intercept_, sgd_clf<span style="color: #666666">.</span>coef_)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Compute the slope and bias of each decision boundary</span>
|
||||
w1 <span style="color: #666666">=</span> <span style="color: #666666">-</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
b1 <span style="color: #666666">=</span> <span style="color: #666666">-</span>lin_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
w2 <span style="color: #666666">=</span> <span style="color: #666666">-</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
b2 <span style="color: #666666">=</span> <span style="color: #666666">-</span>svm_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
w3 <span style="color: #666666">=</span> <span style="color: #666666">-</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
b3 <span style="color: #666666">=</span> <span style="color: #666666">-</span>sgd_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Transform the decision boundary lines back to the original scale</span>
|
||||
line1 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w1 <span style="color: #666666">+</span> b1], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w1 <span style="color: #666666">+</span> b1]])
|
||||
line2 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w2 <span style="color: #666666">+</span> b2], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w2 <span style="color: #666666">+</span> b2]])
|
||||
line3 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w3 <span style="color: #666666">+</span> b3], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w3 <span style="color: #666666">+</span> b3]])
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Plot all three decision boundaries</span>
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
plt<span style="color: #666666">.</span>plot(line1[:, <span style="color: #666666">0</span>], line1[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">"k:"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"LinearSVC"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(line2[:, <span style="color: #666666">0</span>], line2[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">"b--"</span>, linewidth<span style="color: #666666">=2</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"SVC"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(line3[:, <span style="color: #666666">0</span>], line3[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">"r-"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"SGDClassifier"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==1</span>], <span style="color: #BA2121">"bs"</span>) <span style="color: #408080; font-style: italic"># label="Iris-Versicolor"</span>
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==0</span>], <span style="color: #BA2121">"yo"</span>) <span style="color: #408080; font-style: italic"># label="Iris-Setosa"</span>
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">"Petal length"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">"Petal width"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">"upper center"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>, <span style="color: #666666">5.5</span>, <span style="color: #666666">0</span>, <span style="color: #666666">2</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -255,7 +203,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week46-bs010.html">11</a></li>
|
||||
<li><a href="._week46-bs011.html">12</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs003.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Support Vector Machines, overarching aims', 2, None, '___sec0'),
|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec3'),
|
||||
('The two-dimensional case', 2, None, '___sec4'),
|
||||
('Getting into the details', 2, None, '___sec5'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec6'),
|
||||
('Solving the equations', 2, None, '___sec7'),
|
||||
('Code Example', 2, None, '___sec8'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec9'),
|
||||
('A better approach', 2, None, '___sec10'),
|
||||
'sections': [('Overview of week 46', 2, None, '___sec0'),
|
||||
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
|
||||
('Hyperplanes and all that', 2, None, '___sec2'),
|
||||
('What is a hyperplane?', 2, None, '___sec3'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec4'),
|
||||
('The two-dimensional case', 2, None, '___sec5'),
|
||||
('Getting into the details', 2, None, '___sec6'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec7'),
|
||||
('Solving the equations', 2, None, '___sec8'),
|
||||
('Code Example', 2, None, '___sec9'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec10'),
|
||||
('A better approach', 2, None, '___sec11'),
|
||||
('A quick Reminder on Lagrangian Multipliers',
|
||||
2,
|
||||
None,
|
||||
'___sec11'),
|
||||
('Adding the Multiplier', 2, None, '___sec12'),
|
||||
('Setting up the Problem', 2, None, '___sec13'),
|
||||
('The problem to solve', 2, None, '___sec14'),
|
||||
('The last steps', 2, None, '___sec15'),
|
||||
('A soft classifier', 2, None, '___sec16'),
|
||||
('Soft optmization problem', 2, None, '___sec17'),
|
||||
('Kernels and non-linearity', 2, None, '___sec18'),
|
||||
('The equations', 2, None, '___sec19'),
|
||||
('The problem to solve', 2, None, '___sec20'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec21'),
|
||||
('The moons example', 2, None, '___sec22'),
|
||||
'___sec12'),
|
||||
('Adding the Multiplier', 2, None, '___sec13'),
|
||||
('Setting up the Problem', 2, None, '___sec14'),
|
||||
('The problem to solve', 2, None, '___sec15'),
|
||||
('The last steps', 2, None, '___sec16'),
|
||||
('A soft classifier', 2, None, '___sec17'),
|
||||
('Soft optmization problem', 2, None, '___sec18'),
|
||||
('Kernels and non-linearity', 2, None, '___sec19'),
|
||||
('The equations', 2, None, '___sec20'),
|
||||
('The problem to solve', 2, None, '___sec21'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
|
||||
('The moons example', 2, None, '___sec23'),
|
||||
('Mathematical optimization of convex functions',
|
||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('How do we solve these problems?', 2, None, '___sec24'),
|
||||
('A simple example', 2, None, '___sec25'),
|
||||
('Back to the more realistic cases', 2, None, '___sec26')]}
|
||||
'___sec24'),
|
||||
('How do we solve these problems?', 2, None, '___sec25'),
|
||||
('A simple example', 2, None, '___sec26'),
|
||||
('Back to the more realistic cases', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
||||
<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,34 +155,90 @@ MathJax.Hub.Config({
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||||
<a name="part0003"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec2" class="anchor">What is a hyperplane? </h2>
|
||||
<h2 id="___sec2" class="anchor">Hyperplanes and all that </h2>
|
||||
|
||||
<p>
|
||||
The aim of the SVM algorithm is to find a hyperplane in a
|
||||
\( p \)-dimensional space, where \( p \) is the number of features that
|
||||
distinctly classifies the data points.
|
||||
The theory behind support vector machines (SVM hereafter) is based on
|
||||
the mathematical description of so-called hyperplanes. Let us start
|
||||
with a two-dimensional case. This will also allow us to introduce our
|
||||
first SVM examples. These will be tailored to the case of two specific
|
||||
classes, as displayed in the figure here based on the usage of the petal data.
|
||||
|
||||
<p>
|
||||
In a \( p \)-dimensional space, a hyperplane is what we call an affine subspace of dimension of \( p-1 \).
|
||||
As an example, in two dimension, a hyperplane is simply as straight line while in three dimensions it is
|
||||
a two-dimensional subspace, or stated simply, a plane.
|
||||
|
||||
We assume here that our data set can be well separated into two
|
||||
domains, where a straight line does the job in the separating the two
|
||||
classes. Here the two classes are represented by either squares or
|
||||
circles.
|
||||
<p>
|
||||
In two dimensions, with the variables \( x_1 \) and \( x_2 \), the hyperplane is defined as
|
||||
$$
|
||||
b+w_1x_1+w_2x_2=0,
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( b \) is the intercept and \( w_1 \) and \( w_2 \) define the elements of a vector orthogonal to the line
|
||||
\( b+w_1x_1+w_2x_2=0 \).
|
||||
In two dimensions we define the vectors \( \boldsymbol{x} =[x1,x2] \) and \( \boldsymbol{w}=[w1,w2] \).
|
||||
We can then rewrite the above equation as
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC, LinearSVC
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> SGDClassifier
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'axes.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">14</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'xtick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'ytick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
|
||||
$$
|
||||
\boldsymbol{x}^T\boldsymbol{w}+b=0.
|
||||
$$
|
||||
|
||||
iris <span style="color: #666666">=</span> datasets<span style="color: #666666">.</span>load_iris()
|
||||
X <span style="color: #666666">=</span> iris[<span style="color: #BA2121">"data"</span>][:, (<span style="color: #666666">2</span>, <span style="color: #666666">3</span>)] <span style="color: #408080; font-style: italic"># petal length, petal width</span>
|
||||
y <span style="color: #666666">=</span> iris[<span style="color: #BA2121">"target"</span>]
|
||||
|
||||
setosa_or_versicolor <span style="color: #666666">=</span> (y <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">|</span> (y <span style="color: #666666">==</span> <span style="color: #666666">1</span>)
|
||||
X <span style="color: #666666">=</span> X[setosa_or_versicolor]
|
||||
y <span style="color: #666666">=</span> y[setosa_or_versicolor]
|
||||
|
||||
|
||||
|
||||
C <span style="color: #666666">=</span> <span style="color: #666666">5</span>
|
||||
alpha <span style="color: #666666">=</span> <span style="color: #666666">1</span> <span style="color: #666666">/</span> (C <span style="color: #666666">*</span> <span style="color: #008000">len</span>(X))
|
||||
|
||||
lin_clf <span style="color: #666666">=</span> LinearSVC(loss<span style="color: #666666">=</span><span style="color: #BA2121">"hinge"</span>, C<span style="color: #666666">=</span>C, random_state<span style="color: #666666">=42</span>)
|
||||
svm_clf <span style="color: #666666">=</span> SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"linear"</span>, C<span style="color: #666666">=</span>C)
|
||||
sgd_clf <span style="color: #666666">=</span> SGDClassifier(loss<span style="color: #666666">=</span><span style="color: #BA2121">"hinge"</span>, learning_rate<span style="color: #666666">=</span><span style="color: #BA2121">"constant"</span>, eta0<span style="color: #666666">=0.001</span>, alpha<span style="color: #666666">=</span>alpha,
|
||||
max_iter<span style="color: #666666">=100000</span>, random_state<span style="color: #666666">=42</span>)
|
||||
|
||||
scaler <span style="color: #666666">=</span> StandardScaler()
|
||||
X_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>fit_transform(X)
|
||||
|
||||
lin_clf<span style="color: #666666">.</span>fit(X_scaled, y)
|
||||
svm_clf<span style="color: #666666">.</span>fit(X_scaled, y)
|
||||
sgd_clf<span style="color: #666666">.</span>fit(X_scaled, y)
|
||||
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"LinearSVC: "</span>, lin_clf<span style="color: #666666">.</span>intercept_, lin_clf<span style="color: #666666">.</span>coef_)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"SVC: "</span>, svm_clf<span style="color: #666666">.</span>intercept_, svm_clf<span style="color: #666666">.</span>coef_)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"SGDClassifier(alpha=</span><span style="color: #BB6688; font-weight: bold">{:.5f}</span><span style="color: #BA2121">):"</span><span style="color: #666666">.</span>format(sgd_clf<span style="color: #666666">.</span>alpha), sgd_clf<span style="color: #666666">.</span>intercept_, sgd_clf<span style="color: #666666">.</span>coef_)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Compute the slope and bias of each decision boundary</span>
|
||||
w1 <span style="color: #666666">=</span> <span style="color: #666666">-</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
b1 <span style="color: #666666">=</span> <span style="color: #666666">-</span>lin_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
w2 <span style="color: #666666">=</span> <span style="color: #666666">-</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
b2 <span style="color: #666666">=</span> <span style="color: #666666">-</span>svm_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
w3 <span style="color: #666666">=</span> <span style="color: #666666">-</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
b3 <span style="color: #666666">=</span> <span style="color: #666666">-</span>sgd_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Transform the decision boundary lines back to the original scale</span>
|
||||
line1 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w1 <span style="color: #666666">+</span> b1], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w1 <span style="color: #666666">+</span> b1]])
|
||||
line2 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w2 <span style="color: #666666">+</span> b2], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w2 <span style="color: #666666">+</span> b2]])
|
||||
line3 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w3 <span style="color: #666666">+</span> b3], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w3 <span style="color: #666666">+</span> b3]])
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Plot all three decision boundaries</span>
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
plt<span style="color: #666666">.</span>plot(line1[:, <span style="color: #666666">0</span>], line1[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">"k:"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"LinearSVC"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(line2[:, <span style="color: #666666">0</span>], line2[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">"b--"</span>, linewidth<span style="color: #666666">=2</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"SVC"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(line3[:, <span style="color: #666666">0</span>], line3[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">"r-"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"SGDClassifier"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==1</span>], <span style="color: #BA2121">"bs"</span>) <span style="color: #408080; font-style: italic"># label="Iris-Versicolor"</span>
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==0</span>], <span style="color: #BA2121">"yo"</span>) <span style="color: #408080; font-style: italic"># label="Iris-Setosa"</span>
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">"Petal length"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">"Petal width"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">"upper center"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>, <span style="color: #666666">5.5</span>, <span style="color: #666666">0</span>, <span style="color: #666666">2</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -200,7 +258,7 @@ $$
|
||||
<li><a href="._week46-bs011.html">12</a></li>
|
||||
<li><a href="._week46-bs012.html">13</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs004.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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|
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|
||||
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|
||||
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|
||||
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|
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|
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|
||||
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|
||||
('Back to the more realistic cases', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
||||
<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,44 +155,33 @@ MathJax.Hub.Config({
|
||||
<a name="part0004"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec3" class="anchor">A \( p \)-dimensional space of features </h2>
|
||||
<h2 id="___sec3" class="anchor">What is a hyperplane? </h2>
|
||||
|
||||
<p>
|
||||
We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \).
|
||||
In a \( p \)-dimensional space of say \( p \) features we have a hyperplane defines as
|
||||
$$
|
||||
b+wx_1+w_2x_2+\dots +w_px_p=0.
|
||||
$$
|
||||
|
||||
If we define a
|
||||
matrix \( \boldsymbol{X}=\left[\boldsymbol{x}_1,\boldsymbol{x}_2,\dots, \boldsymbol{x}_p\right] \)
|
||||
of dimension \( n\times p \), where \( n \) represents the observations for each feature and each vector \( x_i \) is a column vector of the matrix \( \boldsymbol{X} \),
|
||||
$$
|
||||
\boldsymbol{x}_i = \begin{bmatrix} x_{i1} \\ x_{i2} \\ \dots \\ \dots \\ x_{ip} \end{bmatrix}.
|
||||
$$
|
||||
|
||||
If the above condition is not met for a given vector \( \boldsymbol{x}_i \) we have
|
||||
$$
|
||||
b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} >0,
|
||||
$$
|
||||
|
||||
if our output \( y_i=1 \).
|
||||
In this case we say that \( \boldsymbol{x}_i \) lies on one of the sides of the hyperplane and if
|
||||
$$
|
||||
b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} < 0,
|
||||
$$
|
||||
|
||||
for the class of observations \( y_i=-1 \),
|
||||
then \( \boldsymbol{x}_i \) lies on the other side.
|
||||
The aim of the SVM algorithm is to find a hyperplane in a
|
||||
\( p \)-dimensional space, where \( p \) is the number of features that
|
||||
distinctly classifies the data points.
|
||||
|
||||
<p>
|
||||
Equivalently, for the two classes of observations we have
|
||||
In a \( p \)-dimensional space, a hyperplane is what we call an affine subspace of dimension of \( p-1 \).
|
||||
As an example, in two dimension, a hyperplane is simply as straight line while in three dimensions it is
|
||||
a two-dimensional subspace, or stated simply, a plane.
|
||||
|
||||
<p>
|
||||
In two dimensions, with the variables \( x_1 \) and \( x_2 \), the hyperplane is defined as
|
||||
$$
|
||||
y_i\left(b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip}\right) > 0.
|
||||
b+w_1x_1+w_2x_2=0,
|
||||
$$
|
||||
|
||||
<p>
|
||||
When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located.
|
||||
where \( b \) is the intercept and \( w_1 \) and \( w_2 \) define the elements of a vector orthogonal to the line
|
||||
\( b+w_1x_1+w_2x_2=0 \).
|
||||
In two dimensions we define the vectors \( \boldsymbol{x} =[x1,x2] \) and \( \boldsymbol{w}=[w1,w2] \).
|
||||
We can then rewrite the above equation as
|
||||
|
||||
$$
|
||||
\boldsymbol{x}^T\boldsymbol{w}+b=0.
|
||||
$$
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -212,7 +203,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a
|
||||
<li><a href="._week46-bs012.html">13</a></li>
|
||||
<li><a href="._week46-bs013.html">14</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs005.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Support Vector Machines, overarching aims', 2, None, '___sec0'),
|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec3'),
|
||||
('The two-dimensional case', 2, None, '___sec4'),
|
||||
('Getting into the details', 2, None, '___sec5'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec6'),
|
||||
('Solving the equations', 2, None, '___sec7'),
|
||||
('Code Example', 2, None, '___sec8'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec9'),
|
||||
('A better approach', 2, None, '___sec10'),
|
||||
'sections': [('Overview of week 46', 2, None, '___sec0'),
|
||||
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
|
||||
('Hyperplanes and all that', 2, None, '___sec2'),
|
||||
('What is a hyperplane?', 2, None, '___sec3'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec4'),
|
||||
('The two-dimensional case', 2, None, '___sec5'),
|
||||
('Getting into the details', 2, None, '___sec6'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec7'),
|
||||
('Solving the equations', 2, None, '___sec8'),
|
||||
('Code Example', 2, None, '___sec9'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec10'),
|
||||
('A better approach', 2, None, '___sec11'),
|
||||
('A quick Reminder on Lagrangian Multipliers',
|
||||
2,
|
||||
None,
|
||||
'___sec11'),
|
||||
('Adding the Multiplier', 2, None, '___sec12'),
|
||||
('Setting up the Problem', 2, None, '___sec13'),
|
||||
('The problem to solve', 2, None, '___sec14'),
|
||||
('The last steps', 2, None, '___sec15'),
|
||||
('A soft classifier', 2, None, '___sec16'),
|
||||
('Soft optmization problem', 2, None, '___sec17'),
|
||||
('Kernels and non-linearity', 2, None, '___sec18'),
|
||||
('The equations', 2, None, '___sec19'),
|
||||
('The problem to solve', 2, None, '___sec20'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec21'),
|
||||
('The moons example', 2, None, '___sec22'),
|
||||
'___sec12'),
|
||||
('Adding the Multiplier', 2, None, '___sec13'),
|
||||
('Setting up the Problem', 2, None, '___sec14'),
|
||||
('The problem to solve', 2, None, '___sec15'),
|
||||
('The last steps', 2, None, '___sec16'),
|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
|
||||
('The moons example', 2, None, '___sec23'),
|
||||
('Mathematical optimization of convex functions',
|
||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('How do we solve these problems?', 2, None, '___sec24'),
|
||||
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|
||||
('Back to the more realistic cases', 2, None, '___sec26')]}
|
||||
'___sec24'),
|
||||
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|
||||
('A simple example', 2, None, '___sec26'),
|
||||
('Back to the more realistic cases', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
||||
<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -151,31 +153,46 @@ MathJax.Hub.Config({
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0005"></a>
|
||||
<!-- !split -->
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec4" class="anchor">The two-dimensional case </h2>
|
||||
<h2 id="___sec4" class="anchor">A \( p \)-dimensional space of features </h2>
|
||||
|
||||
<p>
|
||||
Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional
|
||||
plane. To separate the two classes of data points, there are many
|
||||
possible lines (hyperplanes if you prefer a more strict naming)
|
||||
that could be chosen. Our objective is to find a
|
||||
plane that has the maximum margin, i.e the maximum distance between
|
||||
data points of both classes. Maximizing the margin distance provides
|
||||
some reinforcement so that future data points can be classified with
|
||||
more confidence.
|
||||
We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \).
|
||||
In a \( p \)-dimensional space of say \( p \) features we have a hyperplane defines as
|
||||
$$
|
||||
b+wx_1+w_2x_2+\dots +w_px_p=0.
|
||||
$$
|
||||
|
||||
If we define a
|
||||
matrix \( \boldsymbol{X}=\left[\boldsymbol{x}_1,\boldsymbol{x}_2,\dots, \boldsymbol{x}_p\right] \)
|
||||
of dimension \( n\times p \), where \( n \) represents the observations for each feature and each vector \( x_i \) is a column vector of the matrix \( \boldsymbol{X} \),
|
||||
$$
|
||||
\boldsymbol{x}_i = \begin{bmatrix} x_{i1} \\ x_{i2} \\ \dots \\ \dots \\ x_{ip} \end{bmatrix}.
|
||||
$$
|
||||
|
||||
If the above condition is not met for a given vector \( \boldsymbol{x}_i \) we have
|
||||
$$
|
||||
b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} >0,
|
||||
$$
|
||||
|
||||
if our output \( y_i=1 \).
|
||||
In this case we say that \( \boldsymbol{x}_i \) lies on one of the sides of the hyperplane and if
|
||||
$$
|
||||
b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} < 0,
|
||||
$$
|
||||
|
||||
for the class of observations \( y_i=-1 \),
|
||||
then \( \boldsymbol{x}_i \) lies on the other side.
|
||||
|
||||
<p>
|
||||
What a linear classifier attempts to accomplish is to split the
|
||||
feature space into two half spaces by placing a hyperplane between the
|
||||
data points. This hyperplane will be our decision boundary. All
|
||||
points on one side of the plane will belong to class one and all points
|
||||
on the other side of the plane will belong to the second class two.
|
||||
Equivalently, for the two classes of observations we have
|
||||
$$
|
||||
y_i\left(b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip}\right) > 0.
|
||||
$$
|
||||
|
||||
<p>
|
||||
Unfortunately there are many ways in which we can place a hyperplane
|
||||
to divide the data. Below is an example of two candidate hyperplanes
|
||||
for our data sample.
|
||||
When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -198,7 +215,7 @@ for our data sample.
|
||||
<li><a href="._week46-bs013.html">14</a></li>
|
||||
<li><a href="._week46-bs014.html">15</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs006.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
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<meta name="description" content="Week 46: Support Vector Machines">
|
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<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
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<title>Week 46: Support Vector Machines</title>
|
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<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
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<body>
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<span class="icon-bar"></span>
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<span class="icon-bar"></span>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
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</div>
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@@ -111,33 +112,34 @@ MathJax.Hub.Config({
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<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -151,26 +153,31 @@ MathJax.Hub.Config({
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0006"></a>
|
||||
<!-- !split -->
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec5" class="anchor">Getting into the details </h2>
|
||||
<h2 id="___sec5" class="anchor">The two-dimensional case </h2>
|
||||
|
||||
<p>
|
||||
Let us define the function
|
||||
$$
|
||||
f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0,
|
||||
$$
|
||||
|
||||
as the function that determines the line \( L \) that separates two classes (our two features), see the figure here.
|
||||
Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional
|
||||
plane. To separate the two classes of data points, there are many
|
||||
possible lines (hyperplanes if you prefer a more strict naming)
|
||||
that could be chosen. Our objective is to find a
|
||||
plane that has the maximum margin, i.e the maximum distance between
|
||||
data points of both classes. Maximizing the margin distance provides
|
||||
some reinforcement so that future data points can be classified with
|
||||
more confidence.
|
||||
|
||||
<p>
|
||||
Any point defined by \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_2 \) on the line \( L \) will satisfy \( \boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0 \).
|
||||
What a linear classifier attempts to accomplish is to split the
|
||||
feature space into two half spaces by placing a hyperplane between the
|
||||
data points. This hyperplane will be our decision boundary. All
|
||||
points on one side of the plane will belong to class one and all points
|
||||
on the other side of the plane will belong to the second class two.
|
||||
|
||||
<p>
|
||||
The signed distance \( \delta \) from any point defined by a vector \( \boldsymbol{x} \) and a point \( \boldsymbol{x}_0 \) on the line \( L \) is then
|
||||
$$
|
||||
\delta = \frac{1}{\vert\vert \boldsymbol{w}\vert\vert}(\boldsymbol{w}^T\boldsymbol{x}+b).
|
||||
$$
|
||||
Unfortunately there are many ways in which we can place a hyperplane
|
||||
to divide the data. Below is an example of two candidate hyperplanes
|
||||
for our data sample.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -194,7 +201,7 @@ $$
|
||||
<li><a href="._week46-bs014.html">15</a></li>
|
||||
<li><a href="._week46-bs015.html">16</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs007.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
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<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
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|
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<!-- Bootstrap style: bootstrap -->
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@@ -103,7 +104,7 @@ MathJax.Hub.Config({
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<span class="icon-bar"></span>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
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</div>
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|
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@@ -111,33 +112,34 @@ MathJax.Hub.Config({
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<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,26 +155,23 @@ MathJax.Hub.Config({
|
||||
<a name="part0007"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec6" class="anchor">First attempt at a minimization approach </h2>
|
||||
<h2 id="___sec6" class="anchor">Getting into the details </h2>
|
||||
|
||||
<p>
|
||||
How do we find the parameter \( b \) and the vector \( \boldsymbol{w} \)? What we could
|
||||
do is to define a cost function which now contains the set of all
|
||||
misclassified points \( M \) and attempt to minimize this function
|
||||
Let us define the function
|
||||
$$
|
||||
f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0,
|
||||
$$
|
||||
|
||||
$$
|
||||
C(\boldsymbol{w},b) = -\sum_{i\in M} y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b).
|
||||
$$
|
||||
as the function that determines the line \( L \) that separates two classes (our two features), see the figure here.
|
||||
|
||||
<p>
|
||||
We could now for example define all values \( y_i =1 \) as misclassified in case we have \( \boldsymbol{w}^T\boldsymbol{x}_i+b < 0 \) and the opposite if we have \( y_i=-1 \). Taking the derivatives gives us
|
||||
$$
|
||||
\frac{\partial C}{\partial b} = -\sum_{i\in M} y_i,
|
||||
$$
|
||||
Any point defined by \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_2 \) on the line \( L \) will satisfy \( \boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0 \).
|
||||
|
||||
and
|
||||
<p>
|
||||
The signed distance \( \delta \) from any point defined by a vector \( \boldsymbol{x} \) and a point \( \boldsymbol{x}_0 \) on the line \( L \) is then
|
||||
$$
|
||||
\frac{\partial C}{\partial \boldsymbol{w}} = -\sum_{i\in M} y_ix_i.
|
||||
\delta = \frac{1}{\vert\vert \boldsymbol{w}\vert\vert}(\boldsymbol{w}^T\boldsymbol{x}+b).
|
||||
$$
|
||||
|
||||
<p>
|
||||
@@ -198,7 +197,7 @@ $$
|
||||
<li><a href="._week46-bs015.html">16</a></li>
|
||||
<li><a href="._week46-bs016.html">17</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs008.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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|
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
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<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
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|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
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<!-- Bootstrap style: bootstrap -->
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
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@@ -111,33 +112,34 @@ MathJax.Hub.Config({
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<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,20 +155,27 @@ MathJax.Hub.Config({
|
||||
<a name="part0008"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec7" class="anchor">Solving the equations </h2>
|
||||
<h2 id="___sec7" class="anchor">First attempt at a minimization approach </h2>
|
||||
|
||||
<p>
|
||||
We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations
|
||||
How do we find the parameter \( b \) and the vector \( \boldsymbol{w} \)? What we could
|
||||
do is to define a cost function which now contains the set of all
|
||||
misclassified points \( M \) and attempt to minimize this function
|
||||
|
||||
$$
|
||||
b \leftarrow b +\eta \frac{\partial C}{\partial b},
|
||||
C(\boldsymbol{w},b) = -\sum_{i\in M} y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b).
|
||||
$$
|
||||
|
||||
and
|
||||
<p>
|
||||
We could now for example define all values \( y_i =1 \) as misclassified in case we have \( \boldsymbol{w}^T\boldsymbol{x}_i+b < 0 \) and the opposite if we have \( y_i=-1 \). Taking the derivatives gives us
|
||||
$$
|
||||
\boldsymbol{w} \leftarrow \boldsymbol{w} +\eta \frac{\partial C}{\partial \boldsymbol{w}},
|
||||
\frac{\partial C}{\partial b} = -\sum_{i\in M} y_i,
|
||||
$$
|
||||
|
||||
where \( \eta \) is our by now well-known learning rate.
|
||||
and
|
||||
$$
|
||||
\frac{\partial C}{\partial \boldsymbol{w}} = -\sum_{i\in M} y_ix_i.
|
||||
$$
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -192,7 +201,7 @@ where \( \eta \) is our by now well-known learning rate.
|
||||
<li><a href="._week46-bs016.html">17</a></li>
|
||||
<li><a href="._week46-bs017.html">18</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs009.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
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<meta name="description" content="Week 46: Support Vector Machines">
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<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
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</div>
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<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,17 +155,21 @@ MathJax.Hub.Config({
|
||||
<a name="part0009"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec8" class="anchor">Code Example </h2>
|
||||
<h2 id="___sec8" class="anchor">Solving the equations </h2>
|
||||
|
||||
<p>
|
||||
The equations we discussed above can be coded rather easily (the
|
||||
framework is similar to what we developed for logistic
|
||||
regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way.
|
||||
<p>
|
||||
We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations
|
||||
$$
|
||||
b \leftarrow b +\eta \frac{\partial C}{\partial b},
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\boldsymbol{w} \leftarrow \boldsymbol{w} +\eta \frac{\partial C}{\partial \boldsymbol{w}},
|
||||
$$
|
||||
|
||||
where \( \eta \) is our by now well-known learning rate.
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -189,7 +195,7 @@ regression). We are going to set up a simple case with two classes only and we w
|
||||
<li><a href="._week46-bs017.html">18</a></li>
|
||||
<li><a href="._week46-bs018.html">19</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs010.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
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<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
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|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
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||||
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||||
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||||
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||||
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||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="#___sec9" style="font-size: 80%;">Code Example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
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</ul>
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</li>
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<a name="part0010"></a>
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<h2 id="___sec9" class="anchor">Problems with the Simpler Approach </h2>
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<h2 id="___sec9" class="anchor">Code Example </h2>
|
||||
|
||||
<p>
|
||||
There are however problems with this approach, although it looks
|
||||
pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.
|
||||
|
||||
The equations we discussed above can be coded rather easily (the
|
||||
framework is similar to what we developed for logistic
|
||||
regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way.
|
||||
<p>
|
||||
For small
|
||||
gaps between the entries, we may also end up needing many iterations
|
||||
before the solutions converge and if the data cannot be separated
|
||||
properly into two distinct classes, we may not experience a converge
|
||||
at all.
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>
|
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</pre></div>
|
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<p>
|
||||
<p>
|
||||
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|
||||
@@ -192,7 +192,7 @@ at all.
|
||||
<li><a href="._week46-bs018.html">19</a></li>
|
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<li><a href="._week46-bs019.html">20</a></li>
|
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<li><a href="">...</a></li>
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<li><a href="._week46-bs027.html">28</a></li>
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<li><a href="._week46-bs028.html">29</a></li>
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<li><a href="._week46-bs011.html">»</a></li>
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</ul>
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@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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<meta name="description" content="Week 46: Support Vector Machines">
|
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<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
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|
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<title>Week 46: Support Vector Machines</title>
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<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
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<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,44 +155,18 @@ MathJax.Hub.Config({
|
||||
<a name="part0011"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec10" class="anchor">A better approach </h2>
|
||||
<h2 id="___sec10" class="anchor">Problems with the Simpler Approach </h2>
|
||||
|
||||
<p>
|
||||
A better approach is rather to try to define a large margin between
|
||||
the two classes (if they are well separated from the beginning).
|
||||
There are however problems with this approach, although it looks
|
||||
pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.
|
||||
|
||||
<p>
|
||||
Thus, we wish to find a margin \( M \) with \( \boldsymbol{w} \) normalized to
|
||||
\( \vert\vert \boldsymbol{w}\vert\vert =1 \) subject to the condition
|
||||
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p.
|
||||
$$
|
||||
|
||||
All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( b \) and \( w_1 \) and \( w_2 \) define this line.
|
||||
|
||||
<p>
|
||||
We seek thus the largest value \( M \) defined by
|
||||
$$
|
||||
\frac{1}{\vert \vert \boldsymbol{w}\vert\vert}y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n,
|
||||
$$
|
||||
|
||||
or just
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M\vert \vert \boldsymbol{w}\vert\vert \hspace{0.1cm}\forall i.
|
||||
$$
|
||||
|
||||
If we scale the equation so that \( \vert \vert \boldsymbol{w}\vert\vert = 1/M \), we have to find the minimum of
|
||||
\( \boldsymbol{w}^T\boldsymbol{w}=\vert \vert \boldsymbol{w}\vert\vert \) (the norm) subject to the condition
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i.
|
||||
$$
|
||||
|
||||
<p>
|
||||
We have thus defined our margin as the invers of the norm of
|
||||
\( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as
|
||||
possible margin \( M \). Before we proceed, we need to remind ourselves
|
||||
about Lagrangian multipliers.
|
||||
For small
|
||||
gaps between the entries, we may also end up needing many iterations
|
||||
before the solutions converge and if the data cannot be separated
|
||||
properly into two distinct classes, we may not experience a converge
|
||||
at all.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -218,7 +194,7 @@ about Lagrangian multipliers.
|
||||
<li><a href="._week46-bs019.html">20</a></li>
|
||||
<li><a href="._week46-bs020.html">21</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs012.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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|
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('What is a hyperplane?', 2, None, '___sec2'),
|
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@@ -103,7 +104,7 @@ MathJax.Hub.Config({
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<span class="icon-bar"></span>
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<span class="icon-bar"></span>
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</button>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
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</div>
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<div class="navbar-collapse collapse navbar-responsive-collapse">
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||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
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||||
<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,52 +155,44 @@ MathJax.Hub.Config({
|
||||
<a name="part0012"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec11" class="anchor">A quick Reminder on Lagrangian Multipliers </h2>
|
||||
<h2 id="___sec11" class="anchor">A better approach </h2>
|
||||
|
||||
<p>
|
||||
Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
|
||||
extreme we have
|
||||
$$
|
||||
df=0.
|
||||
$$
|
||||
|
||||
A necessary and sufficient condition is
|
||||
$$
|
||||
\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0,
|
||||
$$
|
||||
|
||||
due to
|
||||
$$
|
||||
df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz.
|
||||
$$
|
||||
|
||||
In many problems the variables \( x,y,z \) are often subject to constraints (such as those above for the margin)
|
||||
so that they are no longer all independent. It is possible at least in principle to use each
|
||||
constraint to eliminate one variable
|
||||
and to proceed with a new and smaller set of independent varables.
|
||||
A better approach is rather to try to define a large margin between
|
||||
the two classes (if they are well separated from the beginning).
|
||||
|
||||
<p>
|
||||
The use of so-called Lagrangian multipliers is an alternative technique when the elimination
|
||||
of variables is incovenient or undesirable. Assume that we have an equation of constraint on
|
||||
the variables \( x,y,z \)
|
||||
Thus, we wish to find a margin \( M \) with \( \boldsymbol{w} \) normalized to
|
||||
\( \vert\vert \boldsymbol{w}\vert\vert =1 \) subject to the condition
|
||||
|
||||
$$
|
||||
\phi(x,y,z) = 0,
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p.
|
||||
$$
|
||||
|
||||
resulting in
|
||||
All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( b \) and \( w_1 \) and \( w_2 \) define this line.
|
||||
|
||||
<p>
|
||||
We seek thus the largest value \( M \) defined by
|
||||
$$
|
||||
d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0.
|
||||
\frac{1}{\vert \vert \boldsymbol{w}\vert\vert}y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n,
|
||||
$$
|
||||
|
||||
Now we cannot set anymore
|
||||
or just
|
||||
$$
|
||||
\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0,
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M\vert \vert \boldsymbol{w}\vert\vert \hspace{0.1cm}\forall i.
|
||||
$$
|
||||
|
||||
if \( df=0 \) is wanted
|
||||
because there are now only two independent variables! Assume \( x \) and \( y \) are the independent
|
||||
variables.
|
||||
Then \( dz \) is no longer arbitrary.
|
||||
If we scale the equation so that \( \vert \vert \boldsymbol{w}\vert\vert = 1/M \), we have to find the minimum of
|
||||
\( \boldsymbol{w}^T\boldsymbol{w}=\vert \vert \boldsymbol{w}\vert\vert \) (the norm) subject to the condition
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i.
|
||||
$$
|
||||
|
||||
<p>
|
||||
We have thus defined our margin as the invers of the norm of
|
||||
\( \boldsymbol{w} \). We want to minimize the norm in order to have a as large as
|
||||
possible margin \( M \). Before we proceed, we need to remind ourselves
|
||||
about Lagrangian multipliers.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -226,7 +220,7 @@ Then \( dz \) is no longer arbitrary.
|
||||
<li><a href="._week46-bs020.html">21</a></li>
|
||||
<li><a href="._week46-bs021.html">22</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs013.html">»</a></li>
|
||||
</ul>
|
||||
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||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
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<meta name="description" content="Week 46: Support Vector Machines">
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<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
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</div>
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@@ -111,33 +112,34 @@ MathJax.Hub.Config({
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<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,46 +155,53 @@ MathJax.Hub.Config({
|
||||
<a name="part0013"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec12" class="anchor">Adding the Multiplier </h2>
|
||||
<h2 id="___sec12" class="anchor">A quick Reminder on Lagrangian Multipliers </h2>
|
||||
|
||||
<p>
|
||||
However, we can add to
|
||||
Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
|
||||
extreme we have
|
||||
$$
|
||||
df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz,
|
||||
df=0.
|
||||
$$
|
||||
|
||||
a multiplum of \( d\phi \), viz. \( \lambda d\phi \), resulting in
|
||||
A necessary and sufficient condition is
|
||||
$$
|
||||
df+\lambda d\phi = (\frac{\partial f}{\partial z}+\lambda
|
||||
\frac{\partial \phi}{\partial x})dx+(\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y})dy+
|
||||
(\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z})dz =0.
|
||||
\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0,
|
||||
$$
|
||||
|
||||
Our multiplier is chosen so that
|
||||
due to
|
||||
$$
|
||||
\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z} =0.
|
||||
df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz.
|
||||
$$
|
||||
|
||||
In many problems the variables \( x,y,z \) are often subject to constraints (such as those above for the margin)
|
||||
so that they are no longer all independent. It is possible at least in principle to use each
|
||||
constraint to eliminate one variable
|
||||
and to proceed with a new and smaller set of independent varables.
|
||||
|
||||
<p>
|
||||
We need to remember that we took \( dx \) and \( dy \) to be arbitrary and thus we must have
|
||||
The use of so-called Lagrangian multipliers is an alternative technique when the elimination
|
||||
of variables is incovenient or undesirable. Assume that we have an equation of constraint on
|
||||
the variables \( x,y,z \)
|
||||
$$
|
||||
\frac{\partial f}{\partial x}+\lambda\frac{\partial \phi}{\partial x} =0,
|
||||
\phi(x,y,z) = 0,
|
||||
$$
|
||||
|
||||
and
|
||||
resulting in
|
||||
$$
|
||||
\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y} =0.
|
||||
d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0.
|
||||
$$
|
||||
|
||||
When all these equations are satisfied, \( df=0 \). We have four unknowns, \( x,y,z \) and
|
||||
\( \lambda \). Actually we want only \( x,y,z \), \( \lambda \) needs not to be determined,
|
||||
it is therefore often called
|
||||
Lagrange's undetermined multiplier.
|
||||
If we have a set of constraints \( \phi_k \) we have the equations
|
||||
Now we cannot set anymore
|
||||
$$
|
||||
\frac{\partial f}{\partial x_i}+\sum_k\lambda_k\frac{\partial \phi_k}{\partial x_i} =0.
|
||||
\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0,
|
||||
$$
|
||||
|
||||
if \( df=0 \) is wanted
|
||||
because there are now only two independent variables! Assume \( x \) and \( y \) are the independent
|
||||
variables.
|
||||
Then \( dz \) is no longer arbitrary.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -219,7 +228,7 @@ $$
|
||||
<li><a href="._week46-bs021.html">22</a></li>
|
||||
<li><a href="._week46-bs022.html">23</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs014.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
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||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
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<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
|
||||
|
||||
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|
||||
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|
||||
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|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec3'),
|
||||
('The two-dimensional case', 2, None, '___sec4'),
|
||||
('Getting into the details', 2, None, '___sec5'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec6'),
|
||||
('Solving the equations', 2, None, '___sec7'),
|
||||
('Code Example', 2, None, '___sec8'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec9'),
|
||||
('A better approach', 2, None, '___sec10'),
|
||||
'sections': [('Overview of week 46', 2, None, '___sec0'),
|
||||
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
|
||||
('Hyperplanes and all that', 2, None, '___sec2'),
|
||||
('What is a hyperplane?', 2, None, '___sec3'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec4'),
|
||||
('The two-dimensional case', 2, None, '___sec5'),
|
||||
('Getting into the details', 2, None, '___sec6'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec7'),
|
||||
('Solving the equations', 2, None, '___sec8'),
|
||||
('Code Example', 2, None, '___sec9'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec10'),
|
||||
('A better approach', 2, None, '___sec11'),
|
||||
('A quick Reminder on Lagrangian Multipliers',
|
||||
2,
|
||||
None,
|
||||
'___sec11'),
|
||||
('Adding the Multiplier', 2, None, '___sec12'),
|
||||
('Setting up the Problem', 2, None, '___sec13'),
|
||||
('The problem to solve', 2, None, '___sec14'),
|
||||
('The last steps', 2, None, '___sec15'),
|
||||
('A soft classifier', 2, None, '___sec16'),
|
||||
('Soft optmization problem', 2, None, '___sec17'),
|
||||
('Kernels and non-linearity', 2, None, '___sec18'),
|
||||
('The equations', 2, None, '___sec19'),
|
||||
('The problem to solve', 2, None, '___sec20'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec21'),
|
||||
('The moons example', 2, None, '___sec22'),
|
||||
'___sec12'),
|
||||
('Adding the Multiplier', 2, None, '___sec13'),
|
||||
('Setting up the Problem', 2, None, '___sec14'),
|
||||
('The problem to solve', 2, None, '___sec15'),
|
||||
('The last steps', 2, None, '___sec16'),
|
||||
('A soft classifier', 2, None, '___sec17'),
|
||||
('Soft optmization problem', 2, None, '___sec18'),
|
||||
('Kernels and non-linearity', 2, None, '___sec19'),
|
||||
('The equations', 2, None, '___sec20'),
|
||||
('The problem to solve', 2, None, '___sec21'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
|
||||
('The moons example', 2, None, '___sec23'),
|
||||
('Mathematical optimization of convex functions',
|
||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('How do we solve these problems?', 2, None, '___sec24'),
|
||||
('A simple example', 2, None, '___sec25'),
|
||||
('Back to the more realistic cases', 2, None, '___sec26')]}
|
||||
'___sec24'),
|
||||
('How do we solve these problems?', 2, None, '___sec25'),
|
||||
('A simple example', 2, None, '___sec26'),
|
||||
('Back to the more realistic cases', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
||||
<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,43 +155,45 @@ MathJax.Hub.Config({
|
||||
<a name="part0014"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec13" class="anchor">Setting up the Problem </h2>
|
||||
In order to solve the above problem, we define the following Lagrangian function to be minimized
|
||||
$$
|
||||
{\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right],
|
||||
$$
|
||||
|
||||
where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \).
|
||||
<h2 id="___sec13" class="anchor">Adding the Multiplier </h2>
|
||||
|
||||
<p>
|
||||
Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain
|
||||
However, we can add to
|
||||
$$
|
||||
\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0,
|
||||
df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz,
|
||||
$$
|
||||
|
||||
and
|
||||
a multiplum of \( d\phi \), viz. \( \lambda d\phi \), resulting in
|
||||
$$
|
||||
\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i.
|
||||
df+\lambda d\phi = (\frac{\partial f}{\partial z}+\lambda
|
||||
\frac{\partial \phi}{\partial x})dx+(\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y})dy+
|
||||
(\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z})dz =0.
|
||||
$$
|
||||
|
||||
Inserting these constraints into the equation for \( {\cal L} \) we obtain
|
||||
Our multiplier is chosen so that
|
||||
$$
|
||||
{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j,
|
||||
\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z} =0.
|
||||
$$
|
||||
|
||||
subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \).
|
||||
We must in addition satisfy the <a href="https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" target="_self">Karush-Kuhn-Tucker</a> (KKT) condition
|
||||
<p>
|
||||
We need to remember that we took \( dx \) and \( dy \) to be arbitrary and thus we must have
|
||||
$$
|
||||
\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -1\right] \hspace{0.1cm}\forall i.
|
||||
\frac{\partial f}{\partial x}+\lambda\frac{\partial \phi}{\partial x} =0,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y} =0.
|
||||
$$
|
||||
|
||||
<ol>
|
||||
<li> If \( \lambda_i > 0 \), then \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) and we say that \( x_i \) is on the boundary.</li>
|
||||
<li> If \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).</li>
|
||||
</ol>
|
||||
|
||||
When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin \( M \).
|
||||
When all these equations are satisfied, \( df=0 \). We have four unknowns, \( x,y,z \) and
|
||||
\( \lambda \). Actually we want only \( x,y,z \), \( \lambda \) needs not to be determined,
|
||||
it is therefore often called
|
||||
Lagrange's undetermined multiplier.
|
||||
If we have a set of constraints \( \phi_k \) we have the equations
|
||||
$$
|
||||
\frac{\partial f}{\partial x_i}+\sum_k\lambda_k\frac{\partial \phi_k}{\partial x_i} =0.
|
||||
$$
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -217,7 +221,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
|
||||
<li><a href="._week46-bs022.html">23</a></li>
|
||||
<li><a href="._week46-bs023.html">24</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs015.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec3'),
|
||||
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|
||||
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||||
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||||
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|
||||
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
|
||||
('Hyperplanes and all that', 2, None, '___sec2'),
|
||||
('What is a hyperplane?', 2, None, '___sec3'),
|
||||
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|
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|
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|
||||
2,
|
||||
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|
||||
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|
||||
('Adding the Multiplier', 2, None, '___sec12'),
|
||||
('Setting up the Problem', 2, None, '___sec13'),
|
||||
('The problem to solve', 2, None, '___sec14'),
|
||||
('The last steps', 2, None, '___sec15'),
|
||||
('A soft classifier', 2, None, '___sec16'),
|
||||
('Soft optmization problem', 2, None, '___sec17'),
|
||||
('Kernels and non-linearity', 2, None, '___sec18'),
|
||||
('The equations', 2, None, '___sec19'),
|
||||
('The problem to solve', 2, None, '___sec20'),
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("Different kernels and Mercer's theorem", 2, None, '___sec21'),
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('The moons example', 2, None, '___sec22'),
|
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|
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|
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|
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end of tocinfo -->
|
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|
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<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
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<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
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|
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<div class="navbar-collapse collapse navbar-responsive-collapse">
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@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
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<li class="dropdown">
|
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,26 +155,43 @@ MathJax.Hub.Config({
|
||||
<a name="part0015"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec14" class="anchor">The problem to solve </h2>
|
||||
<h2 id="___sec14" class="anchor">Setting up the Problem </h2>
|
||||
In order to solve the above problem, we define the following Lagrangian function to be minimized
|
||||
$$
|
||||
{\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right],
|
||||
$$
|
||||
|
||||
where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \).
|
||||
|
||||
<p>
|
||||
We can rewrite
|
||||
Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain
|
||||
$$
|
||||
\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i.
|
||||
$$
|
||||
|
||||
Inserting these constraints into the equation for \( {\cal L} \) we obtain
|
||||
$$
|
||||
{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j,
|
||||
$$
|
||||
|
||||
and its constraints in terms of a matrix-vector problem where we minimize w.r.t. \( \lambda \) the following problem
|
||||
subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \).
|
||||
We must in addition satisfy the <a href="https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" target="_self">Karush-Kuhn-Tucker</a> (KKT) condition
|
||||
$$
|
||||
\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1\boldsymbol{x}_1^T\boldsymbol{x}_1 & y_1y_2\boldsymbol{x}_1^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_1^T\boldsymbol{x}_n \\
|
||||
y_2y_1\boldsymbol{x}_2^T\boldsymbol{x}_1 & y_2y_2\boldsymbol{x}_2^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_2^T\boldsymbol{x}_n \\
|
||||
\dots & \dots & \dots & \dots & \dots \\
|
||||
\dots & \dots & \dots & \dots & \dots \\
|
||||
y_ny_1\boldsymbol{x}_n^T\boldsymbol{x}_1 & y_ny_2\boldsymbol{x}_n^T\boldsymbol{x}_2 & \dots & \dots & y_ny_n\boldsymbol{x}_n^T\boldsymbol{x}_n \\
|
||||
\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda},
|
||||
\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -1\right] \hspace{0.1cm}\forall i.
|
||||
$$
|
||||
|
||||
subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and
|
||||
\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \).
|
||||
|
||||
<ol>
|
||||
<li> If \( \lambda_i > 0 \), then \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) and we say that \( x_i \) is on the boundary.</li>
|
||||
<li> If \( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).</li>
|
||||
</ol>
|
||||
|
||||
When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin \( M \).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -200,7 +219,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
|
||||
<li><a href="._week46-bs023.html">24</a></li>
|
||||
<li><a href="._week46-bs024.html">25</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs016.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
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|
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<!-- Bootstrap style: bootstrap -->
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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|
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|
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|
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@@ -103,7 +104,7 @@ MathJax.Hub.Config({
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<span class="icon-bar"></span>
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<span class="icon-bar"></span>
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</button>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
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</div>
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@@ -111,33 +112,34 @@ MathJax.Hub.Config({
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<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,36 +155,26 @@ MathJax.Hub.Config({
|
||||
<a name="part0016"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec15" class="anchor">The last steps </h2>
|
||||
<h2 id="___sec15" class="anchor">The problem to solve </h2>
|
||||
|
||||
<p>
|
||||
Solving the above problem, yields the values of \( \lambda_i \).
|
||||
To find the coefficients of your hyperplane we need simply to compute
|
||||
We can rewrite
|
||||
$$
|
||||
\boldsymbol{w}=\sum_{i} \lambda_iy_i\boldsymbol{x}_i.
|
||||
{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j,
|
||||
$$
|
||||
|
||||
With our vector \( \boldsymbol{w} \) we can in turn find the value of the intercept \( b \) (here in two dimensions) via
|
||||
and its constraints in terms of a matrix-vector problem where we minimize w.r.t. \( \lambda \) the following problem
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1,
|
||||
\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1\boldsymbol{x}_1^T\boldsymbol{x}_1 & y_1y_2\boldsymbol{x}_1^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_1^T\boldsymbol{x}_n \\
|
||||
y_2y_1\boldsymbol{x}_2^T\boldsymbol{x}_1 & y_2y_2\boldsymbol{x}_2^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_2^T\boldsymbol{x}_n \\
|
||||
\dots & \dots & \dots & \dots & \dots \\
|
||||
\dots & \dots & \dots & \dots & \dots \\
|
||||
y_ny_1\boldsymbol{x}_n^T\boldsymbol{x}_1 & y_ny_2\boldsymbol{x}_n^T\boldsymbol{x}_2 & \dots & \dots & y_ny_n\boldsymbol{x}_n^T\boldsymbol{x}_n \\
|
||||
\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda},
|
||||
$$
|
||||
|
||||
resulting in
|
||||
$$
|
||||
b = \frac{1}{y_i}-\boldsymbol{w}^T\boldsymbol{x}_i,
|
||||
$$
|
||||
|
||||
or if we write it out in terms of the support vectors only, with \( N_s \) being their number, we have
|
||||
$$
|
||||
b = \frac{1}{N_s}\sum_{j\in N_s}\left(y_j-\sum_{i=1}^n\lambda_iy_i\boldsymbol{x}_i^T\boldsymbol{x}_j\right).
|
||||
$$
|
||||
|
||||
With our hyperplane coefficients we can use our classifier to assign any observation by simply using
|
||||
$$
|
||||
y_i = \mathrm{sign}(\boldsymbol{w}^T\boldsymbol{x}_i+b).
|
||||
$$
|
||||
|
||||
Below we discuss how to find the optimal values of \( \lambda_i \). Before we proceed however, we discuss now the so-called soft classifier.
|
||||
subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and
|
||||
\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -210,7 +202,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
|
||||
<li><a href="._week46-bs024.html">25</a></li>
|
||||
<li><a href="._week46-bs025.html">26</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs017.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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|
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|
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|
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@@ -103,7 +104,7 @@ MathJax.Hub.Config({
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<span class="icon-bar"></span>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
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@@ -111,33 +112,34 @@ MathJax.Hub.Config({
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<li class="dropdown">
|
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,37 +155,36 @@ MathJax.Hub.Config({
|
||||
<a name="part0017"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec16" class="anchor">A soft classifier </h2>
|
||||
<h2 id="___sec16" class="anchor">The last steps </h2>
|
||||
|
||||
<p>
|
||||
Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
|
||||
Solving the above problem, yields the values of \( \lambda_i \).
|
||||
To find the coefficients of your hyperplane we need simply to compute
|
||||
$$
|
||||
\boldsymbol{w}=\sum_{i} \lambda_iy_i\boldsymbol{x}_i.
|
||||
$$
|
||||
|
||||
<p>
|
||||
Suppose now that classes overlap in feature space, as shown in the
|
||||
figure here. One way to deal with this problem before we define the
|
||||
so-called <b>kernel approach</b>, is to allow a kind of slack in the sense
|
||||
that we allow some points to be on the wrong side of the margin.
|
||||
|
||||
<p>
|
||||
We introduce thus the so-called <b>slack</b> variables \( \boldsymbol{\xi} =[\xi_1,x_2,\dots,x_n] \) and
|
||||
modify our previous equation
|
||||
With our vector \( \boldsymbol{w} \) we can in turn find the value of the intercept \( b \) (here in two dimensions) via
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1,
|
||||
$$
|
||||
|
||||
to
|
||||
resulting in
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i,
|
||||
b = \frac{1}{y_i}-\boldsymbol{w}^T\boldsymbol{x}_i,
|
||||
$$
|
||||
|
||||
with the requirement \( \xi_i\geq 0 \). The total violation is now \( \sum_i\xi \).
|
||||
The value \( \xi_i \) in the constraint the last constraint corresponds to the amount by which the prediction
|
||||
\( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) is on the wrong side of its margin. Hence by bounding the sum \( \sum_i \xi_i \),
|
||||
we bound the total amount by which predictions fall on the wrong side of their margins.
|
||||
or if we write it out in terms of the support vectors only, with \( N_s \) being their number, we have
|
||||
$$
|
||||
b = \frac{1}{N_s}\sum_{j\in N_s}\left(y_j-\sum_{i=1}^n\lambda_iy_i\boldsymbol{x}_i^T\boldsymbol{x}_j\right).
|
||||
$$
|
||||
|
||||
<p>
|
||||
Misclassifications occur when \( \xi_i > 1 \). Thus bounding the total sum by some value \( C \) bounds in turn the total number of
|
||||
misclassifications.
|
||||
With our hyperplane coefficients we can use our classifier to assign any observation by simply using
|
||||
$$
|
||||
y_i = \mathrm{sign}(\boldsymbol{w}^T\boldsymbol{x}_i+b).
|
||||
$$
|
||||
|
||||
Below we discuss how to find the optimal values of \( \lambda_i \). Before we proceed however, we discuss now the so-called soft classifier.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -211,7 +212,7 @@ misclassifications.
|
||||
<li><a href="._week46-bs025.html">26</a></li>
|
||||
<li><a href="._week46-bs026.html">27</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs018.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
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<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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<span class="icon-bar"></span>
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||||
<span class="icon-bar"></span>
|
||||
</button>
|
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
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</div>
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<div class="navbar-collapse collapse navbar-responsive-collapse">
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@@ -111,33 +112,34 @@ MathJax.Hub.Config({
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<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,56 +155,37 @@ MathJax.Hub.Config({
|
||||
<a name="part0018"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec17" class="anchor">Soft optmization problem </h2>
|
||||
<h2 id="___sec17" class="anchor">A soft classifier </h2>
|
||||
|
||||
<p>
|
||||
This has in turn the consequences that we change our optmization problem to finding the minimum of
|
||||
$$
|
||||
{\cal L}=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-(1-\xi_)\right]+C\sum_{i=1}^n\xi_i-\sum_{i=1}^n\gamma_i\xi_i,
|
||||
$$
|
||||
|
||||
subject to
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i,
|
||||
$$
|
||||
|
||||
with the requirement \( \xi_i\geq 0 \).
|
||||
Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
|
||||
|
||||
<p>
|
||||
Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain
|
||||
Suppose now that classes overlap in feature space, as shown in the
|
||||
figure here. One way to deal with this problem before we define the
|
||||
so-called <b>kernel approach</b>, is to allow a kind of slack in the sense
|
||||
that we allow some points to be on the wrong side of the margin.
|
||||
|
||||
<p>
|
||||
We introduce thus the so-called <b>slack</b> variables \( \boldsymbol{\xi} =[\xi_1,x_2,\dots,x_n] \) and
|
||||
modify our previous equation
|
||||
$$
|
||||
\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0,
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1,
|
||||
$$
|
||||
|
||||
and
|
||||
to
|
||||
$$
|
||||
\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i,
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i.
|
||||
$$
|
||||
with the requirement \( \xi_i\geq 0 \). The total violation is now \( \sum_i\xi \).
|
||||
The value \( \xi_i \) in the constraint the last constraint corresponds to the amount by which the prediction
|
||||
\( y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1 \) is on the wrong side of its margin. Hence by bounding the sum \( \sum_i \xi_i \),
|
||||
we bound the total amount by which predictions fall on the wrong side of their margins.
|
||||
|
||||
Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before
|
||||
$$
|
||||
{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j,
|
||||
$$
|
||||
|
||||
but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and \( 0\leq\lambda_i \leq C \).
|
||||
We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads
|
||||
$$
|
||||
\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i,
|
||||
$$
|
||||
|
||||
$$
|
||||
\gamma_i\xi_i = 0,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i.
|
||||
$$
|
||||
<p>
|
||||
Misclassifications occur when \( \xi_i > 1 \). Thus bounding the total sum by some value \( C \) bounds in turn the total number of
|
||||
misclassifications.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -229,6 +212,8 @@ $$
|
||||
<li><a href="._week46-bs025.html">26</a></li>
|
||||
<li><a href="._week46-bs026.html">27</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs019.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
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<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
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@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Support Vector Machines, overarching aims', 2, None, '___sec0'),
|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec3'),
|
||||
('The two-dimensional case', 2, None, '___sec4'),
|
||||
('Getting into the details', 2, None, '___sec5'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec6'),
|
||||
('Solving the equations', 2, None, '___sec7'),
|
||||
('Code Example', 2, None, '___sec8'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec9'),
|
||||
('A better approach', 2, None, '___sec10'),
|
||||
'sections': [('Overview of week 46', 2, None, '___sec0'),
|
||||
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
|
||||
('Hyperplanes and all that', 2, None, '___sec2'),
|
||||
('What is a hyperplane?', 2, None, '___sec3'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec4'),
|
||||
('The two-dimensional case', 2, None, '___sec5'),
|
||||
('Getting into the details', 2, None, '___sec6'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec7'),
|
||||
('Solving the equations', 2, None, '___sec8'),
|
||||
('Code Example', 2, None, '___sec9'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec10'),
|
||||
('A better approach', 2, None, '___sec11'),
|
||||
('A quick Reminder on Lagrangian Multipliers',
|
||||
2,
|
||||
None,
|
||||
'___sec11'),
|
||||
('Adding the Multiplier', 2, None, '___sec12'),
|
||||
('Setting up the Problem', 2, None, '___sec13'),
|
||||
('The problem to solve', 2, None, '___sec14'),
|
||||
('The last steps', 2, None, '___sec15'),
|
||||
('A soft classifier', 2, None, '___sec16'),
|
||||
('Soft optmization problem', 2, None, '___sec17'),
|
||||
('Kernels and non-linearity', 2, None, '___sec18'),
|
||||
('The equations', 2, None, '___sec19'),
|
||||
('The problem to solve', 2, None, '___sec20'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec21'),
|
||||
('The moons example', 2, None, '___sec22'),
|
||||
'___sec12'),
|
||||
('Adding the Multiplier', 2, None, '___sec13'),
|
||||
('Setting up the Problem', 2, None, '___sec14'),
|
||||
('The problem to solve', 2, None, '___sec15'),
|
||||
('The last steps', 2, None, '___sec16'),
|
||||
('A soft classifier', 2, None, '___sec17'),
|
||||
('Soft optmization problem', 2, None, '___sec18'),
|
||||
('Kernels and non-linearity', 2, None, '___sec19'),
|
||||
('The equations', 2, None, '___sec20'),
|
||||
('The problem to solve', 2, None, '___sec21'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
|
||||
('The moons example', 2, None, '___sec23'),
|
||||
('Mathematical optimization of convex functions',
|
||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('How do we solve these problems?', 2, None, '___sec24'),
|
||||
('A simple example', 2, None, '___sec25'),
|
||||
('Back to the more realistic cases', 2, None, '___sec26')]}
|
||||
'___sec24'),
|
||||
('How do we solve these problems?', 2, None, '___sec25'),
|
||||
('A simple example', 2, None, '___sec26'),
|
||||
('Back to the more realistic cases', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
||||
<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,76 +155,57 @@ MathJax.Hub.Config({
|
||||
<a name="part0019"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec18" class="anchor">Kernels and non-linearity </h2>
|
||||
<h2 id="___sec18" class="anchor">Soft optmization problem </h2>
|
||||
|
||||
<p>
|
||||
The cases we have studied till now, were all characterized by two classes
|
||||
with a close to linear separability. The classifiers we have described
|
||||
so far find linear boundaries in our input feature space. It is
|
||||
possible to make our procedure more flexible by exploring the feature
|
||||
space using other basis expansions such as higher-order polynomials,
|
||||
wavelets, splines etc.
|
||||
This has in turn the consequences that we change our optmization problem to finding the minimum of
|
||||
$$
|
||||
{\cal L}=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-(1-\xi_)\right]+C\sum_{i=1}^n\xi_i-\sum_{i=1}^n\gamma_i\xi_i,
|
||||
$$
|
||||
|
||||
subject to
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i,
|
||||
$$
|
||||
|
||||
with the requirement \( \xi_i\geq 0 \).
|
||||
|
||||
<p>
|
||||
If our feature space is not easy to separate, as shown in the figure
|
||||
here, we can achieve a better separation by introducing more complex
|
||||
basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to
|
||||
obtain a separation between the classes which is almost linear.
|
||||
Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain
|
||||
$$
|
||||
\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0,
|
||||
$$
|
||||
|
||||
<p>
|
||||
The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that
|
||||
we need to introduce for example a polynomial transformation to a two-dimensional training set.
|
||||
and
|
||||
$$
|
||||
\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i,
|
||||
$$
|
||||
|
||||
<p>
|
||||
and
|
||||
$$
|
||||
\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i.
|
||||
$$
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
||||
Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before
|
||||
$$
|
||||
{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j,
|
||||
$$
|
||||
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">42</span>)
|
||||
but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and \( 0\leq\lambda_i \leq C \).
|
||||
We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads
|
||||
$$
|
||||
\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i,
|
||||
$$
|
||||
|
||||
<span style="color: #408080; font-style: italic"># To plot pretty figures</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'axes.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">14</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'xtick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'ytick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
$$
|
||||
\gamma_i\xi_i = 0,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i.
|
||||
$$
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets
|
||||
|
||||
|
||||
|
||||
X1D <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-4</span>, <span style="color: #666666">4</span>, <span style="color: #666666">9</span>)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
||||
X2D <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[X1D, X1D<span style="color: #666666">**2</span>]
|
||||
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">0</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">121</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">4</span>), <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">5</span>), <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>gca()<span style="color: #666666">.</span>get_yaxis()<span style="color: #666666">.</span>set_ticks([])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">-0.2</span>, <span style="color: #666666">0.2</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">122</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>axvline(x<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X2D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], X2D[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X2D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], X2D[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==1</span>], <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$x_2$"</span>, fontsize<span style="color: #666666">=20</span>, rotation<span style="color: #666666">=0</span>)
|
||||
plt<span style="color: #666666">.</span>gca()<span style="color: #666666">.</span>get_yaxis()<span style="color: #666666">.</span>set_ticks([<span style="color: #666666">0</span>, <span style="color: #666666">4</span>, <span style="color: #666666">8</span>, <span style="color: #666666">12</span>, <span style="color: #666666">16</span>])
|
||||
plt<span style="color: #666666">.</span>plot([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>], [<span style="color: #666666">6.5</span>, <span style="color: #666666">6.5</span>], <span style="color: #BA2121">"r--"</span>, linewidth<span style="color: #666666">=3</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">17</span>])
|
||||
plt<span style="color: #666666">.</span>subplots_adjust(right<span style="color: #666666">=1</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -247,6 +230,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week46-bs025.html">26</a></li>
|
||||
<li><a href="._week46-bs026.html">27</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs020.html">»</a></li>
|
||||
</ul>
|
||||
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||||
|
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@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
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<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
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|
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<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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|
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|
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|
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|
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<body>
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@@ -103,7 +104,7 @@ MathJax.Hub.Config({
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<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
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</button>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
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</div>
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<div class="navbar-collapse collapse navbar-responsive-collapse">
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@@ -111,33 +112,34 @@ MathJax.Hub.Config({
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<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,48 +155,76 @@ MathJax.Hub.Config({
|
||||
<a name="part0020"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec19" class="anchor">The equations </h2>
|
||||
<h2 id="___sec19" class="anchor">Kernels and non-linearity </h2>
|
||||
|
||||
<p>
|
||||
Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)
|
||||
$$
|
||||
z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right).
|
||||
$$
|
||||
The cases we have studied till now, were all characterized by two classes
|
||||
with a close to linear separability. The classifiers we have described
|
||||
so far find linear boundaries in our input feature space. It is
|
||||
possible to make our procedure more flexible by exploring the feature
|
||||
space using other basis expansions such as higher-order polynomials,
|
||||
wavelets, splines etc.
|
||||
|
||||
<p>
|
||||
With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity)
|
||||
$$
|
||||
{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j,
|
||||
$$
|
||||
|
||||
subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i,
|
||||
$$
|
||||
|
||||
from which we also find \( b \).
|
||||
To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as
|
||||
$$
|
||||
K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j).
|
||||
$$
|
||||
|
||||
For the above example, the kernel reads
|
||||
$$
|
||||
K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2.
|
||||
$$
|
||||
If our feature space is not easy to separate, as shown in the figure
|
||||
here, we can achieve a better separation by introducing more complex
|
||||
basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to
|
||||
obtain a separation between the classes which is almost linear.
|
||||
|
||||
<p>
|
||||
We note that this is nothing but the dot product of the two original
|
||||
vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the
|
||||
product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute
|
||||
the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \).
|
||||
The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that
|
||||
we need to introduce for example a polynomial transformation to a two-dimensional training set.
|
||||
|
||||
<p>
|
||||
This leads to the so-called
|
||||
kernel trick and the result leads to the same as if we went through
|
||||
the trouble of performing the transformation
|
||||
\( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
||||
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">42</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># To plot pretty figures</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'axes.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">14</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'xtick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'ytick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets
|
||||
|
||||
|
||||
|
||||
X1D <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-4</span>, <span style="color: #666666">4</span>, <span style="color: #666666">9</span>)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
||||
X2D <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[X1D, X1D<span style="color: #666666">**2</span>]
|
||||
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">0</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">121</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">4</span>), <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">5</span>), <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>gca()<span style="color: #666666">.</span>get_yaxis()<span style="color: #666666">.</span>set_ticks([])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">-0.2</span>, <span style="color: #666666">0.2</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">122</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>axvline(x<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X2D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], X2D[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X2D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], X2D[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==1</span>], <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$x_2$"</span>, fontsize<span style="color: #666666">=20</span>, rotation<span style="color: #666666">=0</span>)
|
||||
plt<span style="color: #666666">.</span>gca()<span style="color: #666666">.</span>get_yaxis()<span style="color: #666666">.</span>set_ticks([<span style="color: #666666">0</span>, <span style="color: #666666">4</span>, <span style="color: #666666">8</span>, <span style="color: #666666">12</span>, <span style="color: #666666">16</span>])
|
||||
plt<span style="color: #666666">.</span>plot([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>], [<span style="color: #666666">6.5</span>, <span style="color: #666666">6.5</span>], <span style="color: #BA2121">"r--"</span>, linewidth<span style="color: #666666">=3</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">17</span>])
|
||||
plt<span style="color: #666666">.</span>subplots_adjust(right<span style="color: #666666">=1</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -218,6 +248,7 @@ the trouble of performing the transformation
|
||||
<li><a href="._week46-bs025.html">26</a></li>
|
||||
<li><a href="._week46-bs026.html">27</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs021.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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||||
@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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|
||||
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|
||||
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|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
||||
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|
||||
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|
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|
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|
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|
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|
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|
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|
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|
||||
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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('Back to the more realistic cases', 2, None, '___sec27')]}
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end of tocinfo -->
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|
||||
<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
||||
<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,38 +155,47 @@ MathJax.Hub.Config({
|
||||
<a name="part0021"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec20" class="anchor">The problem to solve </h2>
|
||||
Using our definition of the kernel We can rewrite again the Lagrangian
|
||||
$$
|
||||
{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j,
|
||||
$$
|
||||
|
||||
subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) in terms of a convex optimization problem
|
||||
$$
|
||||
\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\
|
||||
y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\
|
||||
\dots & \dots & \dots & \dots & \dots \\
|
||||
\dots & \dots & \dots & \dots & \dots \\
|
||||
y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\
|
||||
\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda},
|
||||
$$
|
||||
|
||||
subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and
|
||||
\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \).
|
||||
If we add the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \).
|
||||
<h2 id="___sec20" class="anchor">The equations </h2>
|
||||
|
||||
<p>
|
||||
We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type
|
||||
Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)
|
||||
$$
|
||||
\begin{align*}
|
||||
&\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber
|
||||
&\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f.
|
||||
\end{align*}
|
||||
z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right).
|
||||
$$
|
||||
|
||||
Below we discuss how to solve these equations. Here we note that the matrix \( \boldsymbol{P} \) has matrix elements \( p_{ij}=y_iy_jK(\boldsymbol{x}_i,\boldsymbol{x}_j) \).
|
||||
Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up. The constraint \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \) leads to \( f=0 \) and \( \boldsymbol{A}=\boldsymbol{y} \). How to set up the matrix \( \boldsymbol{G} \) is discussed later. Here note that the inequalities \( 0\leq \lambda_i \leq C \) can be split up into
|
||||
\( 0\leq \lambda_i \) and \( \lambda_i \leq C \). These two inequalities define then the matrix \( \boldsymbol{G} \) and the vector \( \boldsymbol{h} \).
|
||||
<p>
|
||||
With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity)
|
||||
$$
|
||||
{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j,
|
||||
$$
|
||||
|
||||
subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors
|
||||
$$
|
||||
y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i,
|
||||
$$
|
||||
|
||||
from which we also find \( b \).
|
||||
To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as
|
||||
$$
|
||||
K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j).
|
||||
$$
|
||||
|
||||
For the above example, the kernel reads
|
||||
$$
|
||||
K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2.
|
||||
$$
|
||||
|
||||
<p>
|
||||
We note that this is nothing but the dot product of the two original
|
||||
vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the
|
||||
product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute
|
||||
the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \).
|
||||
|
||||
<p>
|
||||
This leads to the so-called
|
||||
kernel trick and the result leads to the same as if we went through
|
||||
the trouble of performing the transformation
|
||||
\( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -208,6 +219,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.
|
||||
<li><a href="._week46-bs025.html">26</a></li>
|
||||
<li><a href="._week46-bs026.html">27</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs022.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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||||
@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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|
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|
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|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
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<span class="icon-bar"></span>
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||||
<span class="icon-bar"></span>
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</button>
|
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
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<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
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<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,40 +155,38 @@ MathJax.Hub.Config({
|
||||
<a name="part0022"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec21" class="anchor">Different kernels and Mercer's theorem </h2>
|
||||
|
||||
<p>
|
||||
There are several popular kernels being used. These are
|
||||
|
||||
<ol>
|
||||
<li> Linear: \( K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y} \),</li>
|
||||
<li> Polynomial: \( K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d \),</li>
|
||||
<li> Gaussian Radial Basis Function: \( K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)} \),</li>
|
||||
<li> Tanh: \( K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)} \),</li>
|
||||
</ol>
|
||||
|
||||
and many other ones.
|
||||
|
||||
<p>
|
||||
An important theorem for us is <a href="https://en.wikipedia.org/wiki/Mercer%27s_theorem" target="_self">Mercer's
|
||||
theorem</a>. The
|
||||
theorem states that if a kernel function \( K \) is symmetric, continuous
|
||||
and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then there
|
||||
exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into
|
||||
another space (possibly with much higher dimensions) such that
|
||||
|
||||
<h2 id="___sec21" class="anchor">The problem to solve </h2>
|
||||
Using our definition of the kernel We can rewrite again the Lagrangian
|
||||
$$
|
||||
K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j).
|
||||
{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j,
|
||||
$$
|
||||
|
||||
<p>
|
||||
So you can use \( K \) as a kernel since you know \( \phi \) exists, even if
|
||||
you don’t know what \( \phi \) is.
|
||||
subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) in terms of a convex optimization problem
|
||||
$$
|
||||
\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\
|
||||
y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\
|
||||
\dots & \dots & \dots & \dots & \dots \\
|
||||
\dots & \dots & \dots & \dots & \dots \\
|
||||
y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\
|
||||
\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda},
|
||||
$$
|
||||
|
||||
subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and
|
||||
\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \).
|
||||
If we add the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \).
|
||||
|
||||
<p>
|
||||
Note that some frequently used kernels (such as the Sigmoid kernel)
|
||||
don’t respect all of Mercer’s conditions, yet they generally work well
|
||||
in practice.
|
||||
We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type
|
||||
$$
|
||||
\begin{align*}
|
||||
&\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber
|
||||
&\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f.
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Below we discuss how to solve these equations. Here we note that the matrix \( \boldsymbol{P} \) has matrix elements \( p_{ij}=y_iy_jK(\boldsymbol{x}_i,\boldsymbol{x}_j) \).
|
||||
Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up. The constraint \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \) leads to \( f=0 \) and \( \boldsymbol{A}=\boldsymbol{y} \). How to set up the matrix \( \boldsymbol{G} \) is discussed later. Here note that the inequalities \( 0\leq \lambda_i \leq C \) can be split up into
|
||||
\( 0\leq \lambda_i \) and \( \lambda_i \leq C \). These two inequalities define then the matrix \( \boldsymbol{G} \) and the vector \( \boldsymbol{h} \).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -209,6 +209,7 @@ in practice.
|
||||
<li><a href="._week46-bs025.html">26</a></li>
|
||||
<li><a href="._week46-bs026.html">27</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs023.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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||||
|
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|
||||
{'highest level': 2,
|
||||
'sections': [('Support Vector Machines, overarching aims', 2, None, '___sec0'),
|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
||||
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|
||||
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end of tocinfo -->
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|
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<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
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<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
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<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,199 +155,41 @@ MathJax.Hub.Config({
|
||||
<a name="part0023"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec22" class="anchor">The moons example </h2>
|
||||
<h2 id="___sec22" class="anchor">Different kernels and Mercer's theorem </h2>
|
||||
|
||||
<p>
|
||||
There are several popular kernels being used. These are
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">__future__</span> <span style="color: #008000; font-weight: bold">import</span> division, print_function, unicode_literals
|
||||
<ol>
|
||||
<li> Linear: \( K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y} \),</li>
|
||||
<li> Polynomial: \( K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d \),</li>
|
||||
<li> Gaussian Radial Basis Function: \( K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)} \),</li>
|
||||
<li> Tanh: \( K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)} \),</li>
|
||||
</ol>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">42</span>)
|
||||
and many other ones.
|
||||
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'axes.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">14</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'xtick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'ytick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
<p>
|
||||
An important theorem for us is <a href="https://en.wikipedia.org/wiki/Mercer%27s_theorem" target="_self">Mercer's
|
||||
theorem</a>. The
|
||||
theorem states that if a kernel function \( K \) is symmetric, continuous
|
||||
and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then there
|
||||
exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into
|
||||
another space (possibly with much higher dimensions) such that
|
||||
|
||||
$$
|
||||
K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j).
|
||||
$$
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets
|
||||
<p>
|
||||
So you can use \( K \) as a kernel since you know \( \phi \) exists, even if
|
||||
you don’t know what \( \phi \) is.
|
||||
|
||||
<p>
|
||||
Note that some frequently used kernels (such as the Sigmoid kernel)
|
||||
don’t respect all of Mercer’s conditions, yet they generally work well
|
||||
in practice.
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> Pipeline
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> LinearSVC
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> make_moons
|
||||
X, y <span style="color: #666666">=</span> make_moons(n_samples<span style="color: #666666">=100</span>, noise<span style="color: #666666">=0.15</span>, random_state<span style="color: #666666">=42</span>)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">plot_dataset</span>(X, y, axes):
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==1</span>], <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>axis(axes)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$x_2$"</span>, fontsize<span style="color: #666666">=20</span>, rotation<span style="color: #666666">=0</span>)
|
||||
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> make_moons
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> Pipeline
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
|
||||
|
||||
polynomial_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"poly_features"</span>, PolynomialFeatures(degree<span style="color: #666666">=3</span>)),
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, LinearSVC(C<span style="color: #666666">=10</span>, loss<span style="color: #666666">=</span><span style="color: #BA2121">"hinge"</span>, random_state<span style="color: #666666">=42</span>))
|
||||
])
|
||||
|
||||
polynomial_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">plot_predictions</span>(clf, axes):
|
||||
x0s <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(axes[<span style="color: #666666">0</span>], axes[<span style="color: #666666">1</span>], <span style="color: #666666">100</span>)
|
||||
x1s <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(axes[<span style="color: #666666">2</span>], axes[<span style="color: #666666">3</span>], <span style="color: #666666">100</span>)
|
||||
x0, x1 <span style="color: #666666">=</span> np<span style="color: #666666">.</span>meshgrid(x0s, x1s)
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[x0<span style="color: #666666">.</span>ravel(), x1<span style="color: #666666">.</span>ravel()]
|
||||
y_pred <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X)<span style="color: #666666">.</span>reshape(x0<span style="color: #666666">.</span>shape)
|
||||
y_decision <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>decision_function(X)<span style="color: #666666">.</span>reshape(x0<span style="color: #666666">.</span>shape)
|
||||
plt<span style="color: #666666">.</span>contourf(x0, x1, y_pred, cmap<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>cm<span style="color: #666666">.</span>brg, alpha<span style="color: #666666">=0.2</span>)
|
||||
plt<span style="color: #666666">.</span>contourf(x0, x1, y_decision, cmap<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>cm<span style="color: #666666">.</span>brg, alpha<span style="color: #666666">=0.1</span>)
|
||||
|
||||
plot_predictions(polynomial_svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
|
||||
poly_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"poly"</span>, degree<span style="color: #666666">=3</span>, coef0<span style="color: #666666">=1</span>, C<span style="color: #666666">=5</span>))
|
||||
])
|
||||
poly_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
|
||||
poly100_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"poly"</span>, degree<span style="color: #666666">=10</span>, coef0<span style="color: #666666">=100</span>, C<span style="color: #666666">=5</span>))
|
||||
])
|
||||
poly100_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">121</span>)
|
||||
plot_predictions(poly_kernel_svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"$d=3, r=1, C=5$"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">122</span>)
|
||||
plot_predictions(poly100_kernel_svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"$d=10, r=100, C=5$"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">gaussian_rbf</span>(x, landmark, gamma):
|
||||
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>gamma <span style="color: #666666">*</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>norm(x <span style="color: #666666">-</span> landmark, axis<span style="color: #666666">=1</span>)<span style="color: #666666">**2</span>)
|
||||
|
||||
gamma <span style="color: #666666">=</span> <span style="color: #666666">0.3</span>
|
||||
|
||||
x1s <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">200</span>)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
||||
x2s <span style="color: #666666">=</span> gaussian_rbf(x1s, <span style="color: #666666">-2</span>, gamma)
|
||||
x3s <span style="color: #666666">=</span> gaussian_rbf(x1s, <span style="color: #666666">1</span>, gamma)
|
||||
|
||||
XK <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[gaussian_rbf(X1D, <span style="color: #666666">-2</span>, gamma), gaussian_rbf(X1D, <span style="color: #666666">1</span>, gamma)]
|
||||
yk <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">0</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">121</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>scatter(x<span style="color: #666666">=</span>[<span style="color: #666666">-2</span>, <span style="color: #666666">1</span>], y<span style="color: #666666">=</span>[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>], s<span style="color: #666666">=150</span>, alpha<span style="color: #666666">=0.5</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">"red"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==0</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">4</span>), <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==1</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">5</span>), <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x1s, x2s, <span style="color: #BA2121">"g--"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x1s, x3s, <span style="color: #BA2121">"b:"</span>)
|
||||
plt<span style="color: #666666">.</span>gca()<span style="color: #666666">.</span>get_yaxis()<span style="color: #666666">.</span>set_ticks([<span style="color: #666666">0</span>, <span style="color: #666666">0.25</span>, <span style="color: #666666">0.5</span>, <span style="color: #666666">0.75</span>, <span style="color: #666666">1</span>])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"Similarity"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>annotate(<span style="color: #BA2121">r'$\mathbf{x}$'</span>,
|
||||
xy<span style="color: #666666">=</span>(X1D[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>], <span style="color: #666666">0</span>),
|
||||
xytext<span style="color: #666666">=</span>(<span style="color: #666666">-0.5</span>, <span style="color: #666666">0.20</span>),
|
||||
ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>,
|
||||
arrowprops<span style="color: #666666">=</span><span style="color: #008000">dict</span>(facecolor<span style="color: #666666">=</span><span style="color: #BA2121">'black'</span>, shrink<span style="color: #666666">=0.1</span>),
|
||||
fontsize<span style="color: #666666">=18</span>,
|
||||
)
|
||||
plt<span style="color: #666666">.</span>text(<span style="color: #666666">-2</span>, <span style="color: #666666">0.9</span>, <span style="color: #BA2121">"$x_2$"</span>, ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>text(<span style="color: #666666">1</span>, <span style="color: #666666">0.9</span>, <span style="color: #BA2121">"$x_3$"</span>, ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">122</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>axvline(x<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(XK[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==0</span>], XK[:, <span style="color: #666666">1</span>][yk<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(XK[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==1</span>], XK[:, <span style="color: #666666">1</span>][yk<span style="color: #666666">==1</span>], <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_2$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$x_3$ "</span>, fontsize<span style="color: #666666">=20</span>, rotation<span style="color: #666666">=0</span>)
|
||||
plt<span style="color: #666666">.</span>annotate(<span style="color: #BA2121">r'$\phi\left(\mathbf{x}\right)$'</span>,
|
||||
xy<span style="color: #666666">=</span>(XK[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>], XK[<span style="color: #666666">3</span>, <span style="color: #666666">1</span>]),
|
||||
xytext<span style="color: #666666">=</span>(<span style="color: #666666">0.65</span>, <span style="color: #666666">0.50</span>),
|
||||
ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>,
|
||||
arrowprops<span style="color: #666666">=</span><span style="color: #008000">dict</span>(facecolor<span style="color: #666666">=</span><span style="color: #BA2121">'black'</span>, shrink<span style="color: #666666">=0.1</span>),
|
||||
fontsize<span style="color: #666666">=18</span>,
|
||||
)
|
||||
plt<span style="color: #666666">.</span>plot([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>], [<span style="color: #666666">0.57</span>, <span style="color: #666666">-0.1</span>], <span style="color: #BA2121">"r--"</span>, linewidth<span style="color: #666666">=3</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>, <span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>subplots_adjust(right<span style="color: #666666">=1</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
|
||||
x1_example <span style="color: #666666">=</span> X1D[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>]
|
||||
<span style="color: #008000; font-weight: bold">for</span> landmark <span style="color: #AA22FF; font-weight: bold">in</span> (<span style="color: #666666">-2</span>, <span style="color: #666666">1</span>):
|
||||
k <span style="color: #666666">=</span> gaussian_rbf(np<span style="color: #666666">.</span>array([[x1_example]]), np<span style="color: #666666">.</span>array([[landmark]]), gamma)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Phi({}, {}) = {}"</span><span style="color: #666666">.</span>format(x1_example, landmark, k))
|
||||
|
||||
rbf_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"rbf"</span>, gamma<span style="color: #666666">=5</span>, C<span style="color: #666666">=0.001</span>))
|
||||
])
|
||||
rbf_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
|
||||
gamma1, gamma2 <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>, <span style="color: #666666">5</span>
|
||||
C1, C2 <span style="color: #666666">=</span> <span style="color: #666666">0.001</span>, <span style="color: #666666">1000</span>
|
||||
hyperparams <span style="color: #666666">=</span> (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
|
||||
|
||||
svm_clfs <span style="color: #666666">=</span> []
|
||||
<span style="color: #008000; font-weight: bold">for</span> gamma, C <span style="color: #AA22FF; font-weight: bold">in</span> hyperparams:
|
||||
rbf_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"rbf"</span>, gamma<span style="color: #666666">=</span>gamma, C<span style="color: #666666">=</span>C))
|
||||
])
|
||||
rbf_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
svm_clfs<span style="color: #666666">.</span>append(rbf_kernel_svm_clf)
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">7</span>))
|
||||
|
||||
<span style="color: #008000; font-weight: bold">for</span> i, svm_clf <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(svm_clfs):
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">221</span> <span style="color: #666666">+</span> i)
|
||||
plot_predictions(svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
gamma, C <span style="color: #666666">=</span> hyperparams[i]
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"$\gamma = {}, C = {}$"</span><span style="color: #666666">.</span>format(gamma, C), fontsize<span style="color: #666666">=16</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -366,6 +210,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week46-bs025.html">26</a></li>
|
||||
<li><a href="._week46-bs026.html">27</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs024.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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||||
|
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||||
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|
||||
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|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
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|
||||
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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('Back to the more realistic cases', 2, None, '___sec27')]}
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end of tocinfo -->
|
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|
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<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
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<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
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<li class="dropdown">
|
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,28 +155,199 @@ MathJax.Hub.Config({
|
||||
<a name="part0024"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec23" class="anchor">Mathematical optimization of convex functions </h2>
|
||||
|
||||
<h2 id="___sec23" class="anchor">The moons example </h2>
|
||||
<p>
|
||||
A mathematical (quadratic) optimization problem, or just optimization problem, has the form
|
||||
$$
|
||||
\begin{align*}
|
||||
&\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber
|
||||
&\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f.
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
subject to some constraints for say a selected set \( i=1,2,\dots, n \).
|
||||
In our case we are optimizing with respect to the Lagrangian multipliers \( \lambda_i \), and the
|
||||
vector \( \boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n] \) is the optimization variable we are dealing with.
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">__future__</span> <span style="color: #008000; font-weight: bold">import</span> division, print_function, unicode_literals
|
||||
|
||||
<p>
|
||||
In our case we are particularly interested in a class of optimization problems called convex optmization problems.
|
||||
In our discussion on gradient descent methods we discussed at length the definition of a convex function.
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">42</span>)
|
||||
|
||||
<p>
|
||||
Convex optimization problems play a central role in applied mathematics and we recommend strongly <a href="http://web.stanford.edu/~boyd/cvxbook/" target="_self">Boyd and Vandenberghe's text on the topics</a>.
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'axes.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">14</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'xtick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'ytick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets
|
||||
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> Pipeline
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> LinearSVC
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> make_moons
|
||||
X, y <span style="color: #666666">=</span> make_moons(n_samples<span style="color: #666666">=100</span>, noise<span style="color: #666666">=0.15</span>, random_state<span style="color: #666666">=42</span>)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">plot_dataset</span>(X, y, axes):
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==1</span>], <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>axis(axes)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$x_2$"</span>, fontsize<span style="color: #666666">=20</span>, rotation<span style="color: #666666">=0</span>)
|
||||
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> make_moons
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> Pipeline
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
|
||||
|
||||
polynomial_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"poly_features"</span>, PolynomialFeatures(degree<span style="color: #666666">=3</span>)),
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, LinearSVC(C<span style="color: #666666">=10</span>, loss<span style="color: #666666">=</span><span style="color: #BA2121">"hinge"</span>, random_state<span style="color: #666666">=42</span>))
|
||||
])
|
||||
|
||||
polynomial_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">plot_predictions</span>(clf, axes):
|
||||
x0s <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(axes[<span style="color: #666666">0</span>], axes[<span style="color: #666666">1</span>], <span style="color: #666666">100</span>)
|
||||
x1s <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(axes[<span style="color: #666666">2</span>], axes[<span style="color: #666666">3</span>], <span style="color: #666666">100</span>)
|
||||
x0, x1 <span style="color: #666666">=</span> np<span style="color: #666666">.</span>meshgrid(x0s, x1s)
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[x0<span style="color: #666666">.</span>ravel(), x1<span style="color: #666666">.</span>ravel()]
|
||||
y_pred <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X)<span style="color: #666666">.</span>reshape(x0<span style="color: #666666">.</span>shape)
|
||||
y_decision <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>decision_function(X)<span style="color: #666666">.</span>reshape(x0<span style="color: #666666">.</span>shape)
|
||||
plt<span style="color: #666666">.</span>contourf(x0, x1, y_pred, cmap<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>cm<span style="color: #666666">.</span>brg, alpha<span style="color: #666666">=0.2</span>)
|
||||
plt<span style="color: #666666">.</span>contourf(x0, x1, y_decision, cmap<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>cm<span style="color: #666666">.</span>brg, alpha<span style="color: #666666">=0.1</span>)
|
||||
|
||||
plot_predictions(polynomial_svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
|
||||
poly_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"poly"</span>, degree<span style="color: #666666">=3</span>, coef0<span style="color: #666666">=1</span>, C<span style="color: #666666">=5</span>))
|
||||
])
|
||||
poly_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
|
||||
poly100_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"poly"</span>, degree<span style="color: #666666">=10</span>, coef0<span style="color: #666666">=100</span>, C<span style="color: #666666">=5</span>))
|
||||
])
|
||||
poly100_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">121</span>)
|
||||
plot_predictions(poly_kernel_svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"$d=3, r=1, C=5$"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">122</span>)
|
||||
plot_predictions(poly100_kernel_svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"$d=10, r=100, C=5$"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">gaussian_rbf</span>(x, landmark, gamma):
|
||||
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>gamma <span style="color: #666666">*</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>norm(x <span style="color: #666666">-</span> landmark, axis<span style="color: #666666">=1</span>)<span style="color: #666666">**2</span>)
|
||||
|
||||
gamma <span style="color: #666666">=</span> <span style="color: #666666">0.3</span>
|
||||
|
||||
x1s <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">200</span>)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
||||
x2s <span style="color: #666666">=</span> gaussian_rbf(x1s, <span style="color: #666666">-2</span>, gamma)
|
||||
x3s <span style="color: #666666">=</span> gaussian_rbf(x1s, <span style="color: #666666">1</span>, gamma)
|
||||
|
||||
XK <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[gaussian_rbf(X1D, <span style="color: #666666">-2</span>, gamma), gaussian_rbf(X1D, <span style="color: #666666">1</span>, gamma)]
|
||||
yk <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">0</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">121</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>scatter(x<span style="color: #666666">=</span>[<span style="color: #666666">-2</span>, <span style="color: #666666">1</span>], y<span style="color: #666666">=</span>[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>], s<span style="color: #666666">=150</span>, alpha<span style="color: #666666">=0.5</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">"red"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==0</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">4</span>), <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==1</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">5</span>), <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x1s, x2s, <span style="color: #BA2121">"g--"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x1s, x3s, <span style="color: #BA2121">"b:"</span>)
|
||||
plt<span style="color: #666666">.</span>gca()<span style="color: #666666">.</span>get_yaxis()<span style="color: #666666">.</span>set_ticks([<span style="color: #666666">0</span>, <span style="color: #666666">0.25</span>, <span style="color: #666666">0.5</span>, <span style="color: #666666">0.75</span>, <span style="color: #666666">1</span>])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"Similarity"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>annotate(<span style="color: #BA2121">r'$\mathbf</span><span style="color: #BB6688; font-weight: bold">{x}</span><span style="color: #BA2121">$'</span>,
|
||||
xy<span style="color: #666666">=</span>(X1D[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>], <span style="color: #666666">0</span>),
|
||||
xytext<span style="color: #666666">=</span>(<span style="color: #666666">-0.5</span>, <span style="color: #666666">0.20</span>),
|
||||
ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>,
|
||||
arrowprops<span style="color: #666666">=</span><span style="color: #008000">dict</span>(facecolor<span style="color: #666666">=</span><span style="color: #BA2121">'black'</span>, shrink<span style="color: #666666">=0.1</span>),
|
||||
fontsize<span style="color: #666666">=18</span>,
|
||||
)
|
||||
plt<span style="color: #666666">.</span>text(<span style="color: #666666">-2</span>, <span style="color: #666666">0.9</span>, <span style="color: #BA2121">"$x_2$"</span>, ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>text(<span style="color: #666666">1</span>, <span style="color: #666666">0.9</span>, <span style="color: #BA2121">"$x_3$"</span>, ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">122</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>axvline(x<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(XK[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==0</span>], XK[:, <span style="color: #666666">1</span>][yk<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(XK[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==1</span>], XK[:, <span style="color: #666666">1</span>][yk<span style="color: #666666">==1</span>], <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_2$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$x_3$ "</span>, fontsize<span style="color: #666666">=20</span>, rotation<span style="color: #666666">=0</span>)
|
||||
plt<span style="color: #666666">.</span>annotate(<span style="color: #BA2121">r'$\phi\left(\mathbf</span><span style="color: #BB6688; font-weight: bold">{x}</span><span style="color: #BA2121">\right)$'</span>,
|
||||
xy<span style="color: #666666">=</span>(XK[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>], XK[<span style="color: #666666">3</span>, <span style="color: #666666">1</span>]),
|
||||
xytext<span style="color: #666666">=</span>(<span style="color: #666666">0.65</span>, <span style="color: #666666">0.50</span>),
|
||||
ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>,
|
||||
arrowprops<span style="color: #666666">=</span><span style="color: #008000">dict</span>(facecolor<span style="color: #666666">=</span><span style="color: #BA2121">'black'</span>, shrink<span style="color: #666666">=0.1</span>),
|
||||
fontsize<span style="color: #666666">=18</span>,
|
||||
)
|
||||
plt<span style="color: #666666">.</span>plot([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>], [<span style="color: #666666">0.57</span>, <span style="color: #666666">-0.1</span>], <span style="color: #BA2121">"r--"</span>, linewidth<span style="color: #666666">=3</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>, <span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>subplots_adjust(right<span style="color: #666666">=1</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
|
||||
x1_example <span style="color: #666666">=</span> X1D[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>]
|
||||
<span style="color: #008000; font-weight: bold">for</span> landmark <span style="color: #AA22FF; font-weight: bold">in</span> (<span style="color: #666666">-2</span>, <span style="color: #666666">1</span>):
|
||||
k <span style="color: #666666">=</span> gaussian_rbf(np<span style="color: #666666">.</span>array([[x1_example]]), np<span style="color: #666666">.</span>array([[landmark]]), gamma)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Phi(</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">, </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">) = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(x1_example, landmark, k))
|
||||
|
||||
rbf_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"rbf"</span>, gamma<span style="color: #666666">=5</span>, C<span style="color: #666666">=0.001</span>))
|
||||
])
|
||||
rbf_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
|
||||
gamma1, gamma2 <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>, <span style="color: #666666">5</span>
|
||||
C1, C2 <span style="color: #666666">=</span> <span style="color: #666666">0.001</span>, <span style="color: #666666">1000</span>
|
||||
hyperparams <span style="color: #666666">=</span> (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
|
||||
|
||||
svm_clfs <span style="color: #666666">=</span> []
|
||||
<span style="color: #008000; font-weight: bold">for</span> gamma, C <span style="color: #AA22FF; font-weight: bold">in</span> hyperparams:
|
||||
rbf_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"rbf"</span>, gamma<span style="color: #666666">=</span>gamma, C<span style="color: #666666">=</span>C))
|
||||
])
|
||||
rbf_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
svm_clfs<span style="color: #666666">.</span>append(rbf_kernel_svm_clf)
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">7</span>))
|
||||
|
||||
<span style="color: #008000; font-weight: bold">for</span> i, svm_clf <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(svm_clfs):
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">221</span> <span style="color: #666666">+</span> i)
|
||||
plot_predictions(svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
gamma, C <span style="color: #666666">=</span> hyperparams[i]
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"$\gamma = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">, C = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">$"</span><span style="color: #666666">.</span>format(gamma, C), fontsize<span style="color: #666666">=16</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -194,6 +367,7 @@ Convex optimization problems play a central role in applied mathematics and we r
|
||||
<li><a href="._week46-bs025.html">26</a></li>
|
||||
<li><a href="._week46-bs026.html">27</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs025.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
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||||
|
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@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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|
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@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
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<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
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</div>
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<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
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<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,28 +155,27 @@ MathJax.Hub.Config({
|
||||
<a name="part0025"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec24" class="anchor">How do we solve these problems? </h2>
|
||||
<h2 id="___sec24" class="anchor">Mathematical optimization of convex functions </h2>
|
||||
|
||||
<p>
|
||||
If we use Python as programming language and wish to venture beyond
|
||||
<b>scikit-learn</b>, <b>tensorflow</b> and similar software which makes our
|
||||
lives so much easier, we need to dive into the wonderful world of
|
||||
quadratic programming. We can, if we wish, solve the minimization
|
||||
problem using say standard gradient methods or conjugate gradient
|
||||
methods. However, these methods tend to exhibit a rather slow
|
||||
converge. So, welcome to the promised land of quadratic programming.
|
||||
A mathematical (quadratic) optimization problem, or just optimization problem, has the form
|
||||
$$
|
||||
\begin{align*}
|
||||
&\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber
|
||||
&\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f.
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
subject to some constraints for say a selected set \( i=1,2,\dots, n \).
|
||||
In our case we are optimizing with respect to the Lagrangian multipliers \( \lambda_i \), and the
|
||||
vector \( \boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n] \) is the optimization variable we are dealing with.
|
||||
|
||||
<p>
|
||||
The functions we need are contained in the quadratic programming package <b>CVXOPT</b> and we need to import it together with <b>numpy</b> as
|
||||
In our case we are particularly interested in a class of optimization problems called convex optmization problems.
|
||||
In our discussion on gradient descent methods we discussed at length the definition of a convex function.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">cvxopt</span>
|
||||
</pre></div>
|
||||
<p>
|
||||
This will make our life much easier. You don't need t write your own optimizer.
|
||||
Convex optimization problems play a central role in applied mathematics and we recommend strongly <a href="http://web.stanford.edu/~boyd/cvxbook/" target="_self">Boyd and Vandenberghe's text on the topics</a>.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -194,6 +195,7 @@ This will make our life much easier. You don't need t write your own optimizer.
|
||||
<li class="active"><a href="._week46-bs025.html">26</a></li>
|
||||
<li><a href="._week46-bs026.html">27</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs026.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
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<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
<!-- Bootstrap style: bootstrap -->
|
||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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|
||||
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|
||||
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|
||||
('A $p$-dimensional space of features', 2, None, '___sec3'),
|
||||
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|
||||
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|
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|
||||
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|
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
('A $p$-dimensional space of features', 2, None, '___sec4'),
|
||||
('The two-dimensional case', 2, None, '___sec5'),
|
||||
('Getting into the details', 2, None, '___sec6'),
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|
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|
||||
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|
||||
('A better approach', 2, None, '___sec11'),
|
||||
('A quick Reminder on Lagrangian Multipliers',
|
||||
2,
|
||||
None,
|
||||
'___sec11'),
|
||||
('Adding the Multiplier', 2, None, '___sec12'),
|
||||
('Setting up the Problem', 2, None, '___sec13'),
|
||||
('The problem to solve', 2, None, '___sec14'),
|
||||
('The last steps', 2, None, '___sec15'),
|
||||
('A soft classifier', 2, None, '___sec16'),
|
||||
('Soft optmization problem', 2, None, '___sec17'),
|
||||
('Kernels and non-linearity', 2, None, '___sec18'),
|
||||
('The equations', 2, None, '___sec19'),
|
||||
('The problem to solve', 2, None, '___sec20'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec21'),
|
||||
('The moons example', 2, None, '___sec22'),
|
||||
'___sec12'),
|
||||
('Adding the Multiplier', 2, None, '___sec13'),
|
||||
('Setting up the Problem', 2, None, '___sec14'),
|
||||
('The problem to solve', 2, None, '___sec15'),
|
||||
('The last steps', 2, None, '___sec16'),
|
||||
('A soft classifier', 2, None, '___sec17'),
|
||||
('Soft optmization problem', 2, None, '___sec18'),
|
||||
('Kernels and non-linearity', 2, None, '___sec19'),
|
||||
('The equations', 2, None, '___sec20'),
|
||||
('The problem to solve', 2, None, '___sec21'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
|
||||
('The moons example', 2, None, '___sec23'),
|
||||
('Mathematical optimization of convex functions',
|
||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('How do we solve these problems?', 2, None, '___sec24'),
|
||||
('A simple example', 2, None, '___sec25'),
|
||||
('Back to the more realistic cases', 2, None, '___sec26')]}
|
||||
'___sec24'),
|
||||
('How do we solve these problems?', 2, None, '___sec25'),
|
||||
('A simple example', 2, None, '___sec26'),
|
||||
('Back to the more realistic cases', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
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<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,71 +155,29 @@ MathJax.Hub.Config({
|
||||
<a name="part0026"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec25" class="anchor">A simple example </h2>
|
||||
<h2 id="___sec25" class="anchor">How do we solve these problems? </h2>
|
||||
|
||||
<p>
|
||||
We remind ourselves about the general problem we want to solve
|
||||
$$
|
||||
\begin{align*}
|
||||
&\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber
|
||||
&\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f.
|
||||
\end{align*}
|
||||
$$
|
||||
If we use Python as programming language and wish to venture beyond
|
||||
<b>scikit-learn</b>, <b>tensorflow</b> and similar software which makes our
|
||||
lives so much easier, we need to dive into the wonderful world of
|
||||
quadratic programming. We can, if we wish, solve the minimization
|
||||
problem using say standard gradient methods or conjugate gradient
|
||||
methods. However, these methods tend to exhibit a rather slow
|
||||
converge. So, welcome to the promised land of quadratic programming.
|
||||
|
||||
<p>
|
||||
Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem
|
||||
$$
|
||||
\begin{align*}
|
||||
&\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber
|
||||
&\mathrm{subject to} \\ \nonumber
|
||||
&x, y \geq 0 \\ \nonumber
|
||||
&x+3y \geq 15 \\ \nonumber
|
||||
&2x+5y \leq 100 \\ \nonumber
|
||||
&3x+4y \leq 80. \\ \nonumber
|
||||
\end{align*}
|
||||
$$
|
||||
The functions we need are contained in the quadratic programming package <b>CVXOPT</b> and we need to import it together with <b>numpy</b> as
|
||||
|
||||
The minimization problem can be rewritten in terms of vectors and matrices as (with \( x \) and \( y \) being the unknowns)
|
||||
$$
|
||||
\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix}3\\ 4 \end{bmatrix}^T \begin{bmatrix}x \\ y \end{bmatrix}.
|
||||
$$
|
||||
|
||||
Similarly, we can now set up the inequalities (we need to change \( \geq \) to \( \leq \) by multiplying with \( -1 \) on bot sides) as the following matrix-vector equation
|
||||
$$
|
||||
\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}.
|
||||
$$
|
||||
|
||||
We have collapsed all the inequalities into a single matrix \( \boldsymbol{G} \). We see also that our matrix
|
||||
$$
|
||||
\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}
|
||||
$$
|
||||
|
||||
is clearly positive semi-definite (all eigenvalues larger or equal zero).
|
||||
Finally, the vector \( \boldsymbol{h} \) is defined as
|
||||
$$
|
||||
\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
Since we don't have any equalities the matrix \( \boldsymbol{A} \) is set to zero
|
||||
The following code solves the equations for us
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Import the necessary packages</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">cvxopt</span> <span style="color: #008000; font-weight: bold">import</span> matrix
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">cvxopt</span> <span style="color: #008000; font-weight: bold">import</span> solvers
|
||||
P <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>diag([<span style="color: #666666">1</span>,<span style="color: #666666">0</span>]), tc<span style="color: #666666">=</span>’d’)
|
||||
q <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>array([<span style="color: #666666">3</span>,<span style="color: #666666">4</span>]), tc<span style="color: #666666">=</span>’d’)
|
||||
G <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>array([[<span style="color: #666666">-1</span>,<span style="color: #666666">0</span>],[<span style="color: #666666">0</span>,<span style="color: #666666">-1</span>],[<span style="color: #666666">-1</span>,<span style="color: #666666">-3</span>],[<span style="color: #666666">2</span>,<span style="color: #666666">5</span>],[<span style="color: #666666">3</span>,<span style="color: #666666">4</span>]]), tc<span style="color: #666666">=</span>’d’)
|
||||
h <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>array([<span style="color: #666666">0</span>,<span style="color: #666666">0</span>,<span style="color: #666666">-15</span>,<span style="color: #666666">100</span>,<span style="color: #666666">80</span>]), tc<span style="color: #666666">=</span>’d’)
|
||||
<span style="color: #408080; font-style: italic"># Construct the QP, invoke solver</span>
|
||||
sol <span style="color: #666666">=</span> solvers<span style="color: #666666">.</span>qp(P,q,G,h)
|
||||
<span style="color: #408080; font-style: italic"># Extract optimal value and solution</span>
|
||||
sol[’x’]
|
||||
sol[’primal objective’]
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">cvxopt</span>
|
||||
</pre></div>
|
||||
<p>
|
||||
This will make our life much easier. You don't need t write your own optimizer.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -235,6 +195,7 @@ sol[’primal objective’]
|
||||
<li><a href="._week46-bs025.html">26</a></li>
|
||||
<li class="active"><a href="._week46-bs026.html">27</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs027.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
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<meta name="description" content="Week 46: Support Vector Machines">
|
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<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
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|
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<title>Week 46: Support Vector Machines</title>
|
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<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
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|
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<!-- Bootstrap style: bootstrap -->
|
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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|
||||
'sections': [('Support Vector Machines, overarching aims', 2, None, '___sec0'),
|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec3'),
|
||||
('The two-dimensional case', 2, None, '___sec4'),
|
||||
('Getting into the details', 2, None, '___sec5'),
|
||||
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|
||||
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|
||||
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|
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|
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||||
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|
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|
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|
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|
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|
||||
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|
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
('Mathematical optimization of convex functions',
|
||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('How do we solve these problems?', 2, None, '___sec24'),
|
||||
('A simple example', 2, None, '___sec25'),
|
||||
('Back to the more realistic cases', 2, None, '___sec26')]}
|
||||
'___sec24'),
|
||||
('How do we solve these problems?', 2, None, '___sec25'),
|
||||
('A simple example', 2, None, '___sec26'),
|
||||
('Back to the more realistic cases', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
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</div>
|
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<div class="navbar-collapse collapse navbar-responsive-collapse">
|
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@@ -111,33 +112,34 @@ MathJax.Hub.Config({
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<li class="dropdown">
|
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
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<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
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<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
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<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -153,28 +155,72 @@ MathJax.Hub.Config({
|
||||
<a name="part0027"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec26" class="anchor">Back to the more realistic cases </h2>
|
||||
<h2 id="___sec26" class="anchor">A simple example </h2>
|
||||
|
||||
<p>
|
||||
We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the <b>slack</b> parameter \( C \) we have
|
||||
We remind ourselves about the general problem we want to solve
|
||||
$$
|
||||
\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\
|
||||
y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2K(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\
|
||||
\dots & \dots & \dots & \dots & \dots \\
|
||||
\dots & \dots & \dots & \dots & \dots \\
|
||||
y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\
|
||||
\end{bmatrix}\boldsymbol{\lambda}-\mathbb{I}\boldsymbol{\lambda},
|
||||
\begin{align*}
|
||||
&\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber
|
||||
&\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f.
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and
|
||||
\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \).
|
||||
With the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \).
|
||||
<p>
|
||||
Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem
|
||||
$$
|
||||
\begin{align*}
|
||||
&\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber
|
||||
&\mathrm{subject to} \\ \nonumber
|
||||
&x, y \geq 0 \\ \nonumber
|
||||
&x+3y \geq 15 \\ \nonumber
|
||||
&2x+5y \leq 100 \\ \nonumber
|
||||
&3x+4y \leq 80. \\ \nonumber
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
The minimization problem can be rewritten in terms of vectors and matrices as (with \( x \) and \( y \) being the unknowns)
|
||||
$$
|
||||
\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix}3\\ 4 \end{bmatrix}^T \begin{bmatrix}x \\ y \end{bmatrix}.
|
||||
$$
|
||||
|
||||
Similarly, we can now set up the inequalities (we need to change \( \geq \) to \( \leq \) by multiplying with \( -1 \) on bot sides) as the following matrix-vector equation
|
||||
$$
|
||||
\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}.
|
||||
$$
|
||||
|
||||
We have collapsed all the inequalities into a single matrix \( \boldsymbol{G} \). We see also that our matrix
|
||||
$$
|
||||
\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}
|
||||
$$
|
||||
|
||||
is clearly positive semi-definite (all eigenvalues larger or equal zero).
|
||||
Finally, the vector \( \boldsymbol{h} \) is defined as
|
||||
$$
|
||||
\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
<b>code will be added</b>
|
||||
|
||||
Since we don't have any equalities the matrix \( \boldsymbol{A} \) is set to zero
|
||||
The following code solves the equations for us
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Import the necessary packages</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">cvxopt</span> <span style="color: #008000; font-weight: bold">import</span> matrix
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">cvxopt</span> <span style="color: #008000; font-weight: bold">import</span> solvers
|
||||
P <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>diag([<span style="color: #666666">1</span>,<span style="color: #666666">0</span>]), tc<span style="color: #666666">=</span>’d’)
|
||||
q <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>array([<span style="color: #666666">3</span>,<span style="color: #666666">4</span>]), tc<span style="color: #666666">=</span>’d’)
|
||||
G <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>array([[<span style="color: #666666">-1</span>,<span style="color: #666666">0</span>],[<span style="color: #666666">0</span>,<span style="color: #666666">-1</span>],[<span style="color: #666666">-1</span>,<span style="color: #666666">-3</span>],[<span style="color: #666666">2</span>,<span style="color: #666666">5</span>],[<span style="color: #666666">3</span>,<span style="color: #666666">4</span>]]), tc<span style="color: #666666">=</span>’d’)
|
||||
h <span style="color: #666666">=</span> matrix(numpy<span style="color: #666666">.</span>array([<span style="color: #666666">0</span>,<span style="color: #666666">0</span>,<span style="color: #666666">-15</span>,<span style="color: #666666">100</span>,<span style="color: #666666">80</span>]), tc<span style="color: #666666">=</span>’d’)
|
||||
<span style="color: #408080; font-style: italic"># Construct the QP, invoke solver</span>
|
||||
sol <span style="color: #666666">=</span> solvers<span style="color: #666666">.</span>qp(P,q,G,h)
|
||||
<span style="color: #408080; font-style: italic"># Extract optimal value and solution</span>
|
||||
sol[’x’]
|
||||
sol[’primal objective’]
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
@@ -190,6 +236,8 @@ With the slack constants this leads to the additional constraint \( 0\leq \lamb
|
||||
<li><a href="._week46-bs025.html">26</a></li>
|
||||
<li><a href="._week46-bs026.html">27</a></li>
|
||||
<li class="active"><a href="._week46-bs027.html">28</a></li>
|
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<li><a href="._week46-bs028.html">29</a></li>
|
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<li><a href="._week46-bs028.html">»</a></li>
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|
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@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
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<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
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|
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<title>Week 46: Support Vector Machines</title>
|
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<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
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<!-- Bootstrap style: bootstrap -->
|
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<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
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@@ -41,39 +41,40 @@ Automatically generated HTML file from DocOnce source
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|
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|
||||
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|
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('How do we solve these problems?', 2, None, '___sec24'),
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'___sec24'),
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('How do we solve these problems?', 2, None, '___sec25'),
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('Back to the more realistic cases', 2, None, '___sec27')]}
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end of tocinfo -->
|
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|
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<body>
|
||||
@@ -103,7 +104,7 @@ MathJax.Hub.Config({
|
||||
<span class="icon-bar"></span>
|
||||
<span class="icon-bar"></span>
|
||||
</button>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
|
||||
<a class="navbar-brand" href="week46-bs.html">Week 46: Gradient Boosting Summary and Support Vector Machines</a>
|
||||
</div>
|
||||
|
||||
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
||||
@@ -111,33 +112,34 @@ MathJax.Hub.Config({
|
||||
<li class="dropdown">
|
||||
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -156,7 +158,7 @@ MathJax.Hub.Config({
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|
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|
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<div class="jumbotron">
|
||||
<center><h1>Week 46: Support Vector Machines</h1></center> <!-- document title -->
|
||||
<center><h1>Week 46: Gradient Boosting Summary and Support Vector Machines</h1></center> <!-- document title -->
|
||||
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||||
<p>
|
||||
<!-- author(s): Morten Hjorth-Jensen -->
|
||||
@@ -172,7 +174,7 @@ MathJax.Hub.Config({
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Sep 16, 2020</h4></center> <!-- date -->
|
||||
<center><h4>Nov 8, 2020</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
|
||||
@@ -196,7 +198,7 @@ MathJax.Hub.Config({
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<li><a href="._week46-bs008.html">9</a></li>
|
||||
<li><a href="._week46-bs009.html">10</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs001.html">»</a></li>
|
||||
</ul>
|
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<!-- ------------------- end of main content --------------- -->
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@@ -3,9 +3,9 @@
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
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<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
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<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
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|
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@@ -132,7 +132,7 @@ MathJax.Hub.Config({
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<center><h1 style="text-align: center;">Week 46: Support Vector Machines</h1></center> <!-- document title -->
|
||||
<center><h1 style="text-align: center;">Week 46: Gradient Boosting Summary and Support Vector Machines</h1></center> <!-- document title -->
|
||||
|
||||
<p>
|
||||
<!-- author(s): Morten Hjorth-Jensen -->
|
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@@ -148,7 +148,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p> <br>
|
||||
<center><h4>Sep 16, 2020</h4></center> <!-- date -->
|
||||
<center><h4>Nov 8, 2020</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
|
||||
@@ -159,7 +159,23 @@ MathJax.Hub.Config({
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec0">Support Vector Machines, overarching aims </h2>
|
||||
<h2 id="___sec0">Overview of week 46 </h2>
|
||||
|
||||
<ul>
|
||||
<p><li> <b>Thursday</b>: Summary of Gradient Boosting and further examples of applications.</li>
|
||||
<p><li> <b>Friday</b>: Support Vector Machines, classification and regression</li>
|
||||
</ul>
|
||||
<p>
|
||||
|
||||
Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) of Hastie et al contains also a good discussion.
|
||||
|
||||
<p>
|
||||
<a href="https://www.youtube.com/watch?v=efR1C6CvhmE&ab_channel=StatQuestwithJoshStarmer" target="_blank">Overview of Support Vector Machines</a>. see also <a href="https://www.youtube.com/watch?v=N1vOgolbjSc&ab_channel=AliceZhao" target="_blank">this video</a>.
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec1">Support Vector Machines, overarching aims </h2>
|
||||
|
||||
<p>
|
||||
A Support Vector Machine (SVM) is a very powerful and versatile
|
||||
@@ -192,7 +208,7 @@ unlikely that we can separate classes easily by say straight lines.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec1">Hyperplanes and all that </h2>
|
||||
<h2 id="___sec2">Hyperplanes and all that </h2>
|
||||
|
||||
<p>
|
||||
The theory behind support vector machines (SVM hereafter) is based on
|
||||
@@ -245,9 +261,9 @@ lin_clf.fit(X_scaled, y)
|
||||
svm_clf.fit(X_scaled, y)
|
||||
sgd_clf.fit(X_scaled, y)
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"LinearSVC: "</span>, lin_clf.intercept_, lin_clf.coef_)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"SVC: "</span>, svm_clf.intercept_, svm_clf.coef_)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"SGDClassifier(alpha={:.5f}):"</span>.format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">"LinearSVC: "</span>, lin_clf.intercept_, lin_clf.coef_)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">"SVC: "</span>, svm_clf.intercept_, svm_clf.coef_)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">"SGDClassifier(alpha={:.5f}):"</span>.format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)
|
||||
|
||||
<span style="color: #228B22"># Compute the slope and bias of each decision boundary</span>
|
||||
w1 = -lin_clf.coef_[<span style="color: #B452CD">0</span>, <span style="color: #B452CD">0</span>]/lin_clf.coef_[<span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span>]
|
||||
@@ -280,7 +296,7 @@ plt.show()
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec2">What is a hyperplane? </h2>
|
||||
<h2 id="___sec3">What is a hyperplane? </h2>
|
||||
|
||||
<p>
|
||||
The aim of the SVM algorithm is to find a hyperplane in a
|
||||
@@ -315,7 +331,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec3">A \( p \)-dimensional space of features </h2>
|
||||
<h2 id="___sec4">A \( p \)-dimensional space of features </h2>
|
||||
|
||||
<p>
|
||||
We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \).
|
||||
@@ -367,7 +383,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec4">The two-dimensional case </h2>
|
||||
<h2 id="___sec5">The two-dimensional case </h2>
|
||||
|
||||
<p>
|
||||
Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional
|
||||
@@ -394,7 +410,7 @@ for our data sample.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec5">Getting into the details </h2>
|
||||
<h2 id="___sec6">Getting into the details </h2>
|
||||
|
||||
<p>
|
||||
Let us define the function
|
||||
@@ -420,7 +436,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec6">First attempt at a minimization approach </h2>
|
||||
<h2 id="___sec7">First attempt at a minimization approach </h2>
|
||||
|
||||
<p>
|
||||
How do we find the parameter \( b \) and the vector \( \boldsymbol{w} \)? What we could
|
||||
@@ -451,7 +467,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec7">Solving the equations </h2>
|
||||
<h2 id="___sec8">Solving the equations </h2>
|
||||
|
||||
<p>
|
||||
We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations
|
||||
@@ -473,7 +489,7 @@ where \( \eta \) is our by now well-known learning rate.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec8">Code Example </h2>
|
||||
<h2 id="___sec9">Code Example </h2>
|
||||
|
||||
<p>
|
||||
The equations we discussed above can be coded rather easily (the
|
||||
@@ -488,7 +504,7 @@ regression). We are going to set up a simple case with two classes only and we w
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec9">Problems with the Simpler Approach </h2>
|
||||
<h2 id="___sec10">Problems with the Simpler Approach </h2>
|
||||
|
||||
<p>
|
||||
There are however problems with this approach, although it looks
|
||||
@@ -504,7 +520,7 @@ at all.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec10">A better approach </h2>
|
||||
<h2 id="___sec11">A better approach </h2>
|
||||
|
||||
<p>
|
||||
A better approach is rather to try to define a large margin between
|
||||
@@ -554,7 +570,7 @@ about Lagrangian multipliers.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec11">A quick Reminder on Lagrangian Multipliers </h2>
|
||||
<h2 id="___sec12">A quick Reminder on Lagrangian Multipliers </h2>
|
||||
|
||||
<p>
|
||||
Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
|
||||
@@ -616,7 +632,7 @@ Then \( dz \) is no longer arbitrary.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec12">Adding the Multiplier </h2>
|
||||
<h2 id="___sec13">Adding the Multiplier </h2>
|
||||
|
||||
<p>
|
||||
However, we can add to
|
||||
@@ -671,7 +687,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec13">Setting up the Problem </h2>
|
||||
<h2 id="___sec14">Setting up the Problem </h2>
|
||||
In order to solve the above problem, we define the following Lagrangian function to be minimized
|
||||
<p> <br>
|
||||
$$
|
||||
@@ -723,7 +739,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec14">The problem to solve </h2>
|
||||
<h2 id="___sec15">The problem to solve </h2>
|
||||
|
||||
<p>
|
||||
We can rewrite
|
||||
@@ -751,7 +767,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec15">The last steps </h2>
|
||||
<h2 id="___sec16">The last steps </h2>
|
||||
|
||||
<p>
|
||||
Solving the above problem, yields the values of \( \lambda_i \).
|
||||
@@ -795,7 +811,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec16">A soft classifier </h2>
|
||||
<h2 id="___sec17">A soft classifier </h2>
|
||||
|
||||
<p>
|
||||
Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
|
||||
@@ -834,7 +850,7 @@ misclassifications.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec17">Soft optmization problem </h2>
|
||||
<h2 id="___sec18">Soft optmization problem </h2>
|
||||
|
||||
<p>
|
||||
This has in turn the consequences that we change our optmization problem to finding the minimum of
|
||||
@@ -906,7 +922,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec18">Kernels and non-linearity </h2>
|
||||
<h2 id="___sec19">Kernels and non-linearity </h2>
|
||||
|
||||
<p>
|
||||
The cases we have studied till now, were all characterized by two classes
|
||||
@@ -954,7 +970,7 @@ y = np.array([<span style="color: #B452CD">0</span>, <span style="color: #B452CD
|
||||
plt.figure(figsize=(<span style="color: #B452CD">11</span>, <span style="color: #B452CD">4</span>))
|
||||
|
||||
plt.subplot(<span style="color: #B452CD">121</span>)
|
||||
plt.grid(<span style="color: #658b00">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.grid(<span style="color: #8B008B; font-weight: bold">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.axhline(y=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.plot(X1D[:, <span style="color: #B452CD">0</span>][y==<span style="color: #B452CD">0</span>], np.zeros(<span style="color: #B452CD">4</span>), <span style="color: #CD5555">"bs"</span>)
|
||||
plt.plot(X1D[:, <span style="color: #B452CD">0</span>][y==<span style="color: #B452CD">1</span>], np.zeros(<span style="color: #B452CD">5</span>), <span style="color: #CD5555">"g^"</span>)
|
||||
@@ -963,7 +979,7 @@ plt.xlabel(<span style="color: #CD5555">r"$x_1$"</span>, fontsize=<spa
|
||||
plt.axis([-<span style="color: #B452CD">4.5</span>, <span style="color: #B452CD">4.5</span>, -<span style="color: #B452CD">0.2</span>, <span style="color: #B452CD">0.2</span>])
|
||||
|
||||
plt.subplot(<span style="color: #B452CD">122</span>)
|
||||
plt.grid(<span style="color: #658b00">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.grid(<span style="color: #8B008B; font-weight: bold">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.axhline(y=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.axvline(x=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.plot(X2D[:, <span style="color: #B452CD">0</span>][y==<span style="color: #B452CD">0</span>], X2D[:, <span style="color: #B452CD">1</span>][y==<span style="color: #B452CD">0</span>], <span style="color: #CD5555">"bs"</span>)
|
||||
@@ -980,7 +996,7 @@ plt.show()
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec19">The equations </h2>
|
||||
<h2 id="___sec20">The equations </h2>
|
||||
|
||||
<p>
|
||||
Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)
|
||||
@@ -1035,7 +1051,7 @@ the trouble of performing the transformation
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec20">The problem to solve </h2>
|
||||
<h2 id="___sec21">The problem to solve </h2>
|
||||
Using our definition of the kernel We can rewrite again the Lagrangian
|
||||
<p> <br>
|
||||
$$
|
||||
@@ -1077,7 +1093,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec21">Different kernels and Mercer's theorem </h2>
|
||||
<h2 id="___sec22">Different kernels and Mercer's theorem </h2>
|
||||
|
||||
<p>
|
||||
There are several popular kernels being used. These are
|
||||
@@ -1118,7 +1134,7 @@ in practice.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec22">The moons example </h2>
|
||||
<h2 id="___sec23">The moons example </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
@@ -1151,7 +1167,7 @@ X, y = make_moons(n_samples=<span style="color: #B452CD">100</span>, noise=<span
|
||||
plt.plot(X[:, <span style="color: #B452CD">0</span>][y==<span style="color: #B452CD">0</span>], X[:, <span style="color: #B452CD">1</span>][y==<span style="color: #B452CD">0</span>], <span style="color: #CD5555">"bs"</span>)
|
||||
plt.plot(X[:, <span style="color: #B452CD">0</span>][y==<span style="color: #B452CD">1</span>], X[:, <span style="color: #B452CD">1</span>][y==<span style="color: #B452CD">1</span>], <span style="color: #CD5555">"g^"</span>)
|
||||
plt.axis(axes)
|
||||
plt.grid(<span style="color: #658b00">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.grid(<span style="color: #8B008B; font-weight: bold">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.xlabel(<span style="color: #CD5555">r"$x_1$"</span>, fontsize=<span style="color: #B452CD">20</span>)
|
||||
plt.ylabel(<span style="color: #CD5555">r"$x_2$"</span>, fontsize=<span style="color: #B452CD">20</span>, rotation=<span style="color: #B452CD">0</span>)
|
||||
|
||||
@@ -1229,7 +1245,7 @@ yk = np.array([<span style="color: #B452CD">0</span>, <span style="color: #B452C
|
||||
plt.figure(figsize=(<span style="color: #B452CD">11</span>, <span style="color: #B452CD">4</span>))
|
||||
|
||||
plt.subplot(<span style="color: #B452CD">121</span>)
|
||||
plt.grid(<span style="color: #658b00">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.grid(<span style="color: #8B008B; font-weight: bold">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.axhline(y=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.scatter(x=[-<span style="color: #B452CD">2</span>, <span style="color: #B452CD">1</span>], y=[<span style="color: #B452CD">0</span>, <span style="color: #B452CD">0</span>], s=<span style="color: #B452CD">150</span>, alpha=<span style="color: #B452CD">0.5</span>, c=<span style="color: #CD5555">"red"</span>)
|
||||
plt.plot(X1D[:, <span style="color: #B452CD">0</span>][yk==<span style="color: #B452CD">0</span>], np.zeros(<span style="color: #B452CD">4</span>), <span style="color: #CD5555">"bs"</span>)
|
||||
@@ -1251,7 +1267,7 @@ plt.text(<span style="color: #B452CD">1</span>, <span style="color: #B452CD">0.9
|
||||
plt.axis([-<span style="color: #B452CD">4.5</span>, <span style="color: #B452CD">4.5</span>, -<span style="color: #B452CD">0.1</span>, <span style="color: #B452CD">1.1</span>])
|
||||
|
||||
plt.subplot(<span style="color: #B452CD">122</span>)
|
||||
plt.grid(<span style="color: #658b00">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.grid(<span style="color: #8B008B; font-weight: bold">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.axhline(y=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.axvline(x=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.plot(XK[:, <span style="color: #B452CD">0</span>][yk==<span style="color: #B452CD">0</span>], XK[:, <span style="color: #B452CD">1</span>][yk==<span style="color: #B452CD">0</span>], <span style="color: #CD5555">"bs"</span>)
|
||||
@@ -1276,7 +1292,7 @@ plt.show()
|
||||
x1_example = X1D[<span style="color: #B452CD">3</span>, <span style="color: #B452CD">0</span>]
|
||||
<span style="color: #8B008B; font-weight: bold">for</span> landmark <span style="color: #8B008B">in</span> (-<span style="color: #B452CD">2</span>, <span style="color: #B452CD">1</span>):
|
||||
k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Phi({}, {}) = {}"</span>.format(x1_example, landmark, k))
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">"Phi({}, {}) = {}"</span>.format(x1_example, landmark, k))
|
||||
|
||||
rbf_kernel_svm_clf = Pipeline([
|
||||
(<span style="color: #CD5555">"scaler"</span>, StandardScaler()),
|
||||
@@ -1315,7 +1331,7 @@ plt.show()
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec23">Mathematical optimization of convex functions </h2>
|
||||
<h2 id="___sec24">Mathematical optimization of convex functions </h2>
|
||||
|
||||
<p>
|
||||
A mathematical (quadratic) optimization problem, or just optimization problem, has the form
|
||||
@@ -1342,7 +1358,7 @@ Convex optimization problems play a central role in applied mathematics and we r
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec24">How do we solve these problems? </h2>
|
||||
<h2 id="___sec25">How do we solve these problems? </h2>
|
||||
|
||||
<p>
|
||||
If we use Python as programming language and wish to venture beyond
|
||||
@@ -1368,7 +1384,7 @@ This will make our life much easier. You don't need t write your own optimizer.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec25">A simple example </h2>
|
||||
<h2 id="___sec26">A simple example </h2>
|
||||
|
||||
<p>
|
||||
We remind ourselves about the general problem we want to solve
|
||||
@@ -1449,7 +1465,7 @@ sol[<span style="color: #a61717; background-color: #e3d2d2">’</span>primal obj
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec26">Back to the more realistic cases </h2>
|
||||
<h2 id="___sec27">Back to the more realistic cases </h2>
|
||||
|
||||
<p>
|
||||
We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the <b>slack</b> parameter \( C \) we have
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
|
||||
<link href="https://cdn.rawgit.com/hplgit/doconce/master/bundled/html_styles/style_solarized_box/css/solarized_light_code.css" rel="stylesheet" type="text/css" title="light"/>
|
||||
@@ -35,39 +35,40 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Support Vector Machines, overarching aims', 2, None, '___sec0'),
|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec3'),
|
||||
('The two-dimensional case', 2, None, '___sec4'),
|
||||
('Getting into the details', 2, None, '___sec5'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec6'),
|
||||
('Solving the equations', 2, None, '___sec7'),
|
||||
('Code Example', 2, None, '___sec8'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec9'),
|
||||
('A better approach', 2, None, '___sec10'),
|
||||
'sections': [('Overview of week 46', 2, None, '___sec0'),
|
||||
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
|
||||
('Hyperplanes and all that', 2, None, '___sec2'),
|
||||
('What is a hyperplane?', 2, None, '___sec3'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec4'),
|
||||
('The two-dimensional case', 2, None, '___sec5'),
|
||||
('Getting into the details', 2, None, '___sec6'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec7'),
|
||||
('Solving the equations', 2, None, '___sec8'),
|
||||
('Code Example', 2, None, '___sec9'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec10'),
|
||||
('A better approach', 2, None, '___sec11'),
|
||||
('A quick Reminder on Lagrangian Multipliers',
|
||||
2,
|
||||
None,
|
||||
'___sec11'),
|
||||
('Adding the Multiplier', 2, None, '___sec12'),
|
||||
('Setting up the Problem', 2, None, '___sec13'),
|
||||
('The problem to solve', 2, None, '___sec14'),
|
||||
('The last steps', 2, None, '___sec15'),
|
||||
('A soft classifier', 2, None, '___sec16'),
|
||||
('Soft optmization problem', 2, None, '___sec17'),
|
||||
('Kernels and non-linearity', 2, None, '___sec18'),
|
||||
('The equations', 2, None, '___sec19'),
|
||||
('The problem to solve', 2, None, '___sec20'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec21'),
|
||||
('The moons example', 2, None, '___sec22'),
|
||||
'___sec12'),
|
||||
('Adding the Multiplier', 2, None, '___sec13'),
|
||||
('Setting up the Problem', 2, None, '___sec14'),
|
||||
('The problem to solve', 2, None, '___sec15'),
|
||||
('The last steps', 2, None, '___sec16'),
|
||||
('A soft classifier', 2, None, '___sec17'),
|
||||
('Soft optmization problem', 2, None, '___sec18'),
|
||||
('Kernels and non-linearity', 2, None, '___sec19'),
|
||||
('The equations', 2, None, '___sec20'),
|
||||
('The problem to solve', 2, None, '___sec21'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
|
||||
('The moons example', 2, None, '___sec23'),
|
||||
('Mathematical optimization of convex functions',
|
||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('How do we solve these problems?', 2, None, '___sec24'),
|
||||
('A simple example', 2, None, '___sec25'),
|
||||
('Back to the more realistic cases', 2, None, '___sec26')]}
|
||||
'___sec24'),
|
||||
('How do we solve these problems?', 2, None, '___sec25'),
|
||||
('A simple example', 2, None, '___sec26'),
|
||||
('Back to the more realistic cases', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -93,7 +94,7 @@ MathJax.Hub.Config({
|
||||
|
||||
|
||||
|
||||
<center><h1>Week 46: Support Vector Machines</h1></center> <!-- document title -->
|
||||
<center><h1>Week 46: Gradient Boosting Summary and Support Vector Machines</h1></center> <!-- document title -->
|
||||
|
||||
<p>
|
||||
<!-- author(s): Morten Hjorth-Jensen -->
|
||||
@@ -109,12 +110,27 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Sep 16, 2020</h4></center> <!-- date -->
|
||||
<center><h4>Nov 8, 2020</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec0">Support Vector Machines, overarching aims </h2>
|
||||
<h2 id="___sec0">Overview of week 46 </h2>
|
||||
|
||||
<ul>
|
||||
<li> <b>Thursday</b>: Summary of Gradient Boosting and further examples of applications.</li>
|
||||
<li> <b>Friday</b>: Support Vector Machines, classification and regression</li>
|
||||
</ul>
|
||||
|
||||
Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) of Hastie et al contains also a good discussion.
|
||||
|
||||
<p>
|
||||
<a href="https://www.youtube.com/watch?v=efR1C6CvhmE&ab_channel=StatQuestwithJoshStarmer" target="_blank">Overview of Support Vector Machines</a>. see also <a href="https://www.youtube.com/watch?v=N1vOgolbjSc&ab_channel=AliceZhao" target="_blank">this video</a>.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec1">Support Vector Machines, overarching aims </h2>
|
||||
|
||||
<p>
|
||||
A Support Vector Machine (SVM) is a very powerful and versatile
|
||||
@@ -147,7 +163,7 @@ unlikely that we can separate classes easily by say straight lines.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec1">Hyperplanes and all that </h2>
|
||||
<h2 id="___sec2">Hyperplanes and all that </h2>
|
||||
|
||||
<p>
|
||||
The theory behind support vector machines (SVM hereafter) is based on
|
||||
@@ -200,9 +216,9 @@ lin_clf.fit(X_scaled, y)
|
||||
svm_clf.fit(X_scaled, y)
|
||||
sgd_clf.fit(X_scaled, y)
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"LinearSVC: "</span>, lin_clf.intercept_, lin_clf.coef_)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"SVC: "</span>, svm_clf.intercept_, svm_clf.coef_)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"SGDClassifier(alpha={:.5f}):"</span>.format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">"LinearSVC: "</span>, lin_clf.intercept_, lin_clf.coef_)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">"SVC: "</span>, svm_clf.intercept_, svm_clf.coef_)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">"SGDClassifier(alpha={:.5f}):"</span>.format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)
|
||||
|
||||
<span style="color: #228B22"># Compute the slope and bias of each decision boundary</span>
|
||||
w1 = -lin_clf.coef_[<span style="color: #B452CD">0</span>, <span style="color: #B452CD">0</span>]/lin_clf.coef_[<span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span>]
|
||||
@@ -234,7 +250,7 @@ plt.show()
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec2">What is a hyperplane? </h2>
|
||||
<h2 id="___sec3">What is a hyperplane? </h2>
|
||||
|
||||
<p>
|
||||
The aim of the SVM algorithm is to find a hyperplane in a
|
||||
@@ -265,7 +281,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec3">A \( p \)-dimensional space of features </h2>
|
||||
<h2 id="___sec4">A \( p \)-dimensional space of features </h2>
|
||||
|
||||
<p>
|
||||
We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \).
|
||||
@@ -307,7 +323,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a
|
||||
<p>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec4">The two-dimensional case </h2>
|
||||
<h2 id="___sec5">The two-dimensional case </h2>
|
||||
|
||||
<p>
|
||||
Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional
|
||||
@@ -334,7 +350,7 @@ for our data sample.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec5">Getting into the details </h2>
|
||||
<h2 id="___sec6">Getting into the details </h2>
|
||||
|
||||
<p>
|
||||
Let us define the function
|
||||
@@ -356,7 +372,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec6">First attempt at a minimization approach </h2>
|
||||
<h2 id="___sec7">First attempt at a minimization approach </h2>
|
||||
|
||||
<p>
|
||||
How do we find the parameter \( b \) and the vector \( \boldsymbol{w} \)? What we could
|
||||
@@ -381,7 +397,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec7">Solving the equations </h2>
|
||||
<h2 id="___sec8">Solving the equations </h2>
|
||||
|
||||
<p>
|
||||
We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations
|
||||
@@ -399,7 +415,7 @@ where \( \eta \) is our by now well-known learning rate.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec8">Code Example </h2>
|
||||
<h2 id="___sec9">Code Example </h2>
|
||||
|
||||
<p>
|
||||
The equations we discussed above can be coded rather easily (the
|
||||
@@ -413,7 +429,7 @@ regression). We are going to set up a simple case with two classes only and we w
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec9">Problems with the Simpler Approach </h2>
|
||||
<h2 id="___sec10">Problems with the Simpler Approach </h2>
|
||||
|
||||
<p>
|
||||
There are however problems with this approach, although it looks
|
||||
@@ -429,7 +445,7 @@ at all.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec10">A better approach </h2>
|
||||
<h2 id="___sec11">A better approach </h2>
|
||||
|
||||
<p>
|
||||
A better approach is rather to try to define a large margin between
|
||||
@@ -471,7 +487,7 @@ about Lagrangian multipliers.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec11">A quick Reminder on Lagrangian Multipliers </h2>
|
||||
<h2 id="___sec12">A quick Reminder on Lagrangian Multipliers </h2>
|
||||
|
||||
<p>
|
||||
Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
|
||||
@@ -521,7 +537,7 @@ Then \( dz \) is no longer arbitrary.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec12">Adding the Multiplier </h2>
|
||||
<h2 id="___sec13">Adding the Multiplier </h2>
|
||||
|
||||
<p>
|
||||
However, we can add to
|
||||
@@ -564,7 +580,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec13">Setting up the Problem </h2>
|
||||
<h2 id="___sec14">Setting up the Problem </h2>
|
||||
In order to solve the above problem, we define the following Lagrangian function to be minimized
|
||||
$$
|
||||
{\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right],
|
||||
@@ -605,7 +621,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec14">The problem to solve </h2>
|
||||
<h2 id="___sec15">The problem to solve </h2>
|
||||
|
||||
<p>
|
||||
We can rewrite
|
||||
@@ -629,7 +645,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec15">The last steps </h2>
|
||||
<h2 id="___sec16">The last steps </h2>
|
||||
|
||||
<p>
|
||||
Solving the above problem, yields the values of \( \lambda_i \).
|
||||
@@ -663,7 +679,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec16">A soft classifier </h2>
|
||||
<h2 id="___sec17">A soft classifier </h2>
|
||||
|
||||
<p>
|
||||
Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
|
||||
@@ -698,7 +714,7 @@ misclassifications.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec17">Soft optmization problem </h2>
|
||||
<h2 id="___sec18">Soft optmization problem </h2>
|
||||
|
||||
<p>
|
||||
This has in turn the consequences that we change our optmization problem to finding the minimum of
|
||||
@@ -752,7 +768,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec18">Kernels and non-linearity </h2>
|
||||
<h2 id="___sec19">Kernels and non-linearity </h2>
|
||||
|
||||
<p>
|
||||
The cases we have studied till now, were all characterized by two classes
|
||||
@@ -800,7 +816,7 @@ y = np.array([<span style="color: #B452CD">0</span>, <span style="color: #B452CD
|
||||
plt.figure(figsize=(<span style="color: #B452CD">11</span>, <span style="color: #B452CD">4</span>))
|
||||
|
||||
plt.subplot(<span style="color: #B452CD">121</span>)
|
||||
plt.grid(<span style="color: #658b00">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.grid(<span style="color: #8B008B; font-weight: bold">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.axhline(y=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.plot(X1D[:, <span style="color: #B452CD">0</span>][y==<span style="color: #B452CD">0</span>], np.zeros(<span style="color: #B452CD">4</span>), <span style="color: #CD5555">"bs"</span>)
|
||||
plt.plot(X1D[:, <span style="color: #B452CD">0</span>][y==<span style="color: #B452CD">1</span>], np.zeros(<span style="color: #B452CD">5</span>), <span style="color: #CD5555">"g^"</span>)
|
||||
@@ -809,7 +825,7 @@ plt.xlabel(<span style="color: #CD5555">r"$x_1$"</span>, fontsize=<spa
|
||||
plt.axis([-<span style="color: #B452CD">4.5</span>, <span style="color: #B452CD">4.5</span>, -<span style="color: #B452CD">0.2</span>, <span style="color: #B452CD">0.2</span>])
|
||||
|
||||
plt.subplot(<span style="color: #B452CD">122</span>)
|
||||
plt.grid(<span style="color: #658b00">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.grid(<span style="color: #8B008B; font-weight: bold">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.axhline(y=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.axvline(x=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.plot(X2D[:, <span style="color: #B452CD">0</span>][y==<span style="color: #B452CD">0</span>], X2D[:, <span style="color: #B452CD">1</span>][y==<span style="color: #B452CD">0</span>], <span style="color: #CD5555">"bs"</span>)
|
||||
@@ -825,7 +841,7 @@ plt.show()
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec19">The equations </h2>
|
||||
<h2 id="___sec20">The equations </h2>
|
||||
|
||||
<p>
|
||||
Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)
|
||||
@@ -870,7 +886,7 @@ the trouble of performing the transformation
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec20">The problem to solve </h2>
|
||||
<h2 id="___sec21">The problem to solve </h2>
|
||||
Using our definition of the kernel We can rewrite again the Lagrangian
|
||||
$$
|
||||
{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j,
|
||||
@@ -906,7 +922,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec21">Different kernels and Mercer's theorem </h2>
|
||||
<h2 id="___sec22">Different kernels and Mercer's theorem </h2>
|
||||
|
||||
<p>
|
||||
There are several popular kernels being used. These are
|
||||
@@ -944,7 +960,7 @@ in practice.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec22">The moons example </h2>
|
||||
<h2 id="___sec23">The moons example </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
@@ -977,7 +993,7 @@ X, y = make_moons(n_samples=<span style="color: #B452CD">100</span>, noise=<span
|
||||
plt.plot(X[:, <span style="color: #B452CD">0</span>][y==<span style="color: #B452CD">0</span>], X[:, <span style="color: #B452CD">1</span>][y==<span style="color: #B452CD">0</span>], <span style="color: #CD5555">"bs"</span>)
|
||||
plt.plot(X[:, <span style="color: #B452CD">0</span>][y==<span style="color: #B452CD">1</span>], X[:, <span style="color: #B452CD">1</span>][y==<span style="color: #B452CD">1</span>], <span style="color: #CD5555">"g^"</span>)
|
||||
plt.axis(axes)
|
||||
plt.grid(<span style="color: #658b00">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.grid(<span style="color: #8B008B; font-weight: bold">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.xlabel(<span style="color: #CD5555">r"$x_1$"</span>, fontsize=<span style="color: #B452CD">20</span>)
|
||||
plt.ylabel(<span style="color: #CD5555">r"$x_2$"</span>, fontsize=<span style="color: #B452CD">20</span>, rotation=<span style="color: #B452CD">0</span>)
|
||||
|
||||
@@ -1055,7 +1071,7 @@ yk = np.array([<span style="color: #B452CD">0</span>, <span style="color: #B452C
|
||||
plt.figure(figsize=(<span style="color: #B452CD">11</span>, <span style="color: #B452CD">4</span>))
|
||||
|
||||
plt.subplot(<span style="color: #B452CD">121</span>)
|
||||
plt.grid(<span style="color: #658b00">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.grid(<span style="color: #8B008B; font-weight: bold">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.axhline(y=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.scatter(x=[-<span style="color: #B452CD">2</span>, <span style="color: #B452CD">1</span>], y=[<span style="color: #B452CD">0</span>, <span style="color: #B452CD">0</span>], s=<span style="color: #B452CD">150</span>, alpha=<span style="color: #B452CD">0.5</span>, c=<span style="color: #CD5555">"red"</span>)
|
||||
plt.plot(X1D[:, <span style="color: #B452CD">0</span>][yk==<span style="color: #B452CD">0</span>], np.zeros(<span style="color: #B452CD">4</span>), <span style="color: #CD5555">"bs"</span>)
|
||||
@@ -1077,7 +1093,7 @@ plt.text(<span style="color: #B452CD">1</span>, <span style="color: #B452CD">0.9
|
||||
plt.axis([-<span style="color: #B452CD">4.5</span>, <span style="color: #B452CD">4.5</span>, -<span style="color: #B452CD">0.1</span>, <span style="color: #B452CD">1.1</span>])
|
||||
|
||||
plt.subplot(<span style="color: #B452CD">122</span>)
|
||||
plt.grid(<span style="color: #658b00">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.grid(<span style="color: #8B008B; font-weight: bold">True</span>, which=<span style="color: #CD5555">'both'</span>)
|
||||
plt.axhline(y=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.axvline(x=<span style="color: #B452CD">0</span>, color=<span style="color: #CD5555">'k'</span>)
|
||||
plt.plot(XK[:, <span style="color: #B452CD">0</span>][yk==<span style="color: #B452CD">0</span>], XK[:, <span style="color: #B452CD">1</span>][yk==<span style="color: #B452CD">0</span>], <span style="color: #CD5555">"bs"</span>)
|
||||
@@ -1102,7 +1118,7 @@ plt.show()
|
||||
x1_example = X1D[<span style="color: #B452CD">3</span>, <span style="color: #B452CD">0</span>]
|
||||
<span style="color: #8B008B; font-weight: bold">for</span> landmark <span style="color: #8B008B">in</span> (-<span style="color: #B452CD">2</span>, <span style="color: #B452CD">1</span>):
|
||||
k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Phi({}, {}) = {}"</span>.format(x1_example, landmark, k))
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">"Phi({}, {}) = {}"</span>.format(x1_example, landmark, k))
|
||||
|
||||
rbf_kernel_svm_clf = Pipeline([
|
||||
(<span style="color: #CD5555">"scaler"</span>, StandardScaler()),
|
||||
@@ -1140,7 +1156,7 @@ plt.show()
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec23">Mathematical optimization of convex functions </h2>
|
||||
<h2 id="___sec24">Mathematical optimization of convex functions </h2>
|
||||
|
||||
<p>
|
||||
A mathematical (quadratic) optimization problem, or just optimization problem, has the form
|
||||
@@ -1165,7 +1181,7 @@ Convex optimization problems play a central role in applied mathematics and we r
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec24">How do we solve these problems? </h2>
|
||||
<h2 id="___sec25">How do we solve these problems? </h2>
|
||||
|
||||
<p>
|
||||
If we use Python as programming language and wish to venture beyond
|
||||
@@ -1191,7 +1207,7 @@ This will make our life much easier. You don't need t write your own optimizer.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec25">A simple example </h2>
|
||||
<h2 id="___sec26">A simple example </h2>
|
||||
|
||||
<p>
|
||||
We remind ourselves about the general problem we want to solve
|
||||
@@ -1259,7 +1275,7 @@ sol[<span style="color: #a61717; background-color: #e3d2d2">’</span>primal obj
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec26">Back to the more realistic cases </h2>
|
||||
<h2 id="___sec27">Back to the more realistic cases </h2>
|
||||
|
||||
<p>
|
||||
We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the <b>slack</b> parameter \( C \) we have
|
||||
|
||||
@@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 46: Support Vector Machines">
|
||||
<meta name="description" content="Week 46: Gradient Boosting Summary and Support Vector Machines">
|
||||
|
||||
<title>Week 46: Support Vector Machines</title>
|
||||
<title>Week 46: Gradient Boosting Summary and Support Vector Machines</title>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
@@ -40,39 +40,40 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Support Vector Machines, overarching aims', 2, None, '___sec0'),
|
||||
('Hyperplanes and all that', 2, None, '___sec1'),
|
||||
('What is a hyperplane?', 2, None, '___sec2'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec3'),
|
||||
('The two-dimensional case', 2, None, '___sec4'),
|
||||
('Getting into the details', 2, None, '___sec5'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec6'),
|
||||
('Solving the equations', 2, None, '___sec7'),
|
||||
('Code Example', 2, None, '___sec8'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec9'),
|
||||
('A better approach', 2, None, '___sec10'),
|
||||
'sections': [('Overview of week 46', 2, None, '___sec0'),
|
||||
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
|
||||
('Hyperplanes and all that', 2, None, '___sec2'),
|
||||
('What is a hyperplane?', 2, None, '___sec3'),
|
||||
('A $p$-dimensional space of features', 2, None, '___sec4'),
|
||||
('The two-dimensional case', 2, None, '___sec5'),
|
||||
('Getting into the details', 2, None, '___sec6'),
|
||||
('First attempt at a minimization approach', 2, None, '___sec7'),
|
||||
('Solving the equations', 2, None, '___sec8'),
|
||||
('Code Example', 2, None, '___sec9'),
|
||||
('Problems with the Simpler Approach', 2, None, '___sec10'),
|
||||
('A better approach', 2, None, '___sec11'),
|
||||
('A quick Reminder on Lagrangian Multipliers',
|
||||
2,
|
||||
None,
|
||||
'___sec11'),
|
||||
('Adding the Multiplier', 2, None, '___sec12'),
|
||||
('Setting up the Problem', 2, None, '___sec13'),
|
||||
('The problem to solve', 2, None, '___sec14'),
|
||||
('The last steps', 2, None, '___sec15'),
|
||||
('A soft classifier', 2, None, '___sec16'),
|
||||
('Soft optmization problem', 2, None, '___sec17'),
|
||||
('Kernels and non-linearity', 2, None, '___sec18'),
|
||||
('The equations', 2, None, '___sec19'),
|
||||
('The problem to solve', 2, None, '___sec20'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec21'),
|
||||
('The moons example', 2, None, '___sec22'),
|
||||
'___sec12'),
|
||||
('Adding the Multiplier', 2, None, '___sec13'),
|
||||
('Setting up the Problem', 2, None, '___sec14'),
|
||||
('The problem to solve', 2, None, '___sec15'),
|
||||
('The last steps', 2, None, '___sec16'),
|
||||
('A soft classifier', 2, None, '___sec17'),
|
||||
('Soft optmization problem', 2, None, '___sec18'),
|
||||
('Kernels and non-linearity', 2, None, '___sec19'),
|
||||
('The equations', 2, None, '___sec20'),
|
||||
('The problem to solve', 2, None, '___sec21'),
|
||||
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
|
||||
('The moons example', 2, None, '___sec23'),
|
||||
('Mathematical optimization of convex functions',
|
||||
2,
|
||||
None,
|
||||
'___sec23'),
|
||||
('How do we solve these problems?', 2, None, '___sec24'),
|
||||
('A simple example', 2, None, '___sec25'),
|
||||
('Back to the more realistic cases', 2, None, '___sec26')]}
|
||||
'___sec24'),
|
||||
('How do we solve these problems?', 2, None, '___sec25'),
|
||||
('A simple example', 2, None, '___sec26'),
|
||||
('Back to the more realistic cases', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -98,7 +99,7 @@ MathJax.Hub.Config({
|
||||
|
||||
|
||||
|
||||
<center><h1>Week 46: Support Vector Machines</h1></center> <!-- document title -->
|
||||
<center><h1>Week 46: Gradient Boosting Summary and Support Vector Machines</h1></center> <!-- document title -->
|
||||
|
||||
<p>
|
||||
<!-- author(s): Morten Hjorth-Jensen -->
|
||||
@@ -114,12 +115,27 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Sep 16, 2020</h4></center> <!-- date -->
|
||||
<center><h4>Nov 8, 2020</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec0">Support Vector Machines, overarching aims </h2>
|
||||
<h2 id="___sec0">Overview of week 46 </h2>
|
||||
|
||||
<ul>
|
||||
<li> <b>Thursday</b>: Summary of Gradient Boosting and further examples of applications.</li>
|
||||
<li> <b>Friday</b>: Support Vector Machines, classification and regression</li>
|
||||
</ul>
|
||||
|
||||
Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) of Hastie et al contains also a good discussion.
|
||||
|
||||
<p>
|
||||
<a href="https://www.youtube.com/watch?v=efR1C6CvhmE&ab_channel=StatQuestwithJoshStarmer" target="_blank">Overview of Support Vector Machines</a>. see also <a href="https://www.youtube.com/watch?v=N1vOgolbjSc&ab_channel=AliceZhao" target="_blank">this video</a>.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec1">Support Vector Machines, overarching aims </h2>
|
||||
|
||||
<p>
|
||||
A Support Vector Machine (SVM) is a very powerful and versatile
|
||||
@@ -152,7 +168,7 @@ unlikely that we can separate classes easily by say straight lines.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec1">Hyperplanes and all that </h2>
|
||||
<h2 id="___sec2">Hyperplanes and all that </h2>
|
||||
|
||||
<p>
|
||||
The theory behind support vector machines (SVM hereafter) is based on
|
||||
@@ -205,9 +221,9 @@ lin_clf<span style="color: #666666">.</span>fit(X_scaled, y)
|
||||
svm_clf<span style="color: #666666">.</span>fit(X_scaled, y)
|
||||
sgd_clf<span style="color: #666666">.</span>fit(X_scaled, y)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"LinearSVC: "</span>, lin_clf<span style="color: #666666">.</span>intercept_, lin_clf<span style="color: #666666">.</span>coef_)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"SVC: "</span>, svm_clf<span style="color: #666666">.</span>intercept_, svm_clf<span style="color: #666666">.</span>coef_)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"SGDClassifier(alpha={:.5f}):"</span><span style="color: #666666">.</span>format(sgd_clf<span style="color: #666666">.</span>alpha), sgd_clf<span style="color: #666666">.</span>intercept_, sgd_clf<span style="color: #666666">.</span>coef_)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"LinearSVC: "</span>, lin_clf<span style="color: #666666">.</span>intercept_, lin_clf<span style="color: #666666">.</span>coef_)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"SVC: "</span>, svm_clf<span style="color: #666666">.</span>intercept_, svm_clf<span style="color: #666666">.</span>coef_)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"SGDClassifier(alpha=</span><span style="color: #BB6688; font-weight: bold">{:.5f}</span><span style="color: #BA2121">):"</span><span style="color: #666666">.</span>format(sgd_clf<span style="color: #666666">.</span>alpha), sgd_clf<span style="color: #666666">.</span>intercept_, sgd_clf<span style="color: #666666">.</span>coef_)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Compute the slope and bias of each decision boundary</span>
|
||||
w1 <span style="color: #666666">=</span> <span style="color: #666666">-</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
|
||||
@@ -239,7 +255,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec2">What is a hyperplane? </h2>
|
||||
<h2 id="___sec3">What is a hyperplane? </h2>
|
||||
|
||||
<p>
|
||||
The aim of the SVM algorithm is to find a hyperplane in a
|
||||
@@ -270,7 +286,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec3">A \( p \)-dimensional space of features </h2>
|
||||
<h2 id="___sec4">A \( p \)-dimensional space of features </h2>
|
||||
|
||||
<p>
|
||||
We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \).
|
||||
@@ -312,7 +328,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a
|
||||
<p>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec4">The two-dimensional case </h2>
|
||||
<h2 id="___sec5">The two-dimensional case </h2>
|
||||
|
||||
<p>
|
||||
Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional
|
||||
@@ -339,7 +355,7 @@ for our data sample.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec5">Getting into the details </h2>
|
||||
<h2 id="___sec6">Getting into the details </h2>
|
||||
|
||||
<p>
|
||||
Let us define the function
|
||||
@@ -361,7 +377,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec6">First attempt at a minimization approach </h2>
|
||||
<h2 id="___sec7">First attempt at a minimization approach </h2>
|
||||
|
||||
<p>
|
||||
How do we find the parameter \( b \) and the vector \( \boldsymbol{w} \)? What we could
|
||||
@@ -386,7 +402,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec7">Solving the equations </h2>
|
||||
<h2 id="___sec8">Solving the equations </h2>
|
||||
|
||||
<p>
|
||||
We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations
|
||||
@@ -404,7 +420,7 @@ where \( \eta \) is our by now well-known learning rate.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec8">Code Example </h2>
|
||||
<h2 id="___sec9">Code Example </h2>
|
||||
|
||||
<p>
|
||||
The equations we discussed above can be coded rather easily (the
|
||||
@@ -418,7 +434,7 @@ regression). We are going to set up a simple case with two classes only and we w
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec9">Problems with the Simpler Approach </h2>
|
||||
<h2 id="___sec10">Problems with the Simpler Approach </h2>
|
||||
|
||||
<p>
|
||||
There are however problems with this approach, although it looks
|
||||
@@ -434,7 +450,7 @@ at all.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec10">A better approach </h2>
|
||||
<h2 id="___sec11">A better approach </h2>
|
||||
|
||||
<p>
|
||||
A better approach is rather to try to define a large margin between
|
||||
@@ -476,7 +492,7 @@ about Lagrangian multipliers.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec11">A quick Reminder on Lagrangian Multipliers </h2>
|
||||
<h2 id="___sec12">A quick Reminder on Lagrangian Multipliers </h2>
|
||||
|
||||
<p>
|
||||
Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
|
||||
@@ -526,7 +542,7 @@ Then \( dz \) is no longer arbitrary.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec12">Adding the Multiplier </h2>
|
||||
<h2 id="___sec13">Adding the Multiplier </h2>
|
||||
|
||||
<p>
|
||||
However, we can add to
|
||||
@@ -569,7 +585,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec13">Setting up the Problem </h2>
|
||||
<h2 id="___sec14">Setting up the Problem </h2>
|
||||
In order to solve the above problem, we define the following Lagrangian function to be minimized
|
||||
$$
|
||||
{\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right],
|
||||
@@ -610,7 +626,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec14">The problem to solve </h2>
|
||||
<h2 id="___sec15">The problem to solve </h2>
|
||||
|
||||
<p>
|
||||
We can rewrite
|
||||
@@ -634,7 +650,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec15">The last steps </h2>
|
||||
<h2 id="___sec16">The last steps </h2>
|
||||
|
||||
<p>
|
||||
Solving the above problem, yields the values of \( \lambda_i \).
|
||||
@@ -668,7 +684,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec16">A soft classifier </h2>
|
||||
<h2 id="___sec17">A soft classifier </h2>
|
||||
|
||||
<p>
|
||||
Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
|
||||
@@ -703,7 +719,7 @@ misclassifications.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec17">Soft optmization problem </h2>
|
||||
<h2 id="___sec18">Soft optmization problem </h2>
|
||||
|
||||
<p>
|
||||
This has in turn the consequences that we change our optmization problem to finding the minimum of
|
||||
@@ -757,7 +773,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec18">Kernels and non-linearity </h2>
|
||||
<h2 id="___sec19">Kernels and non-linearity </h2>
|
||||
|
||||
<p>
|
||||
The cases we have studied till now, were all characterized by two classes
|
||||
@@ -805,7 +821,7 @@ y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>a
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">121</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">4</span>), <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">5</span>), <span style="color: #BA2121">"g^"</span>)
|
||||
@@ -814,7 +830,7 @@ plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r&qu
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">-0.2</span>, <span style="color: #666666">0.2</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">122</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>axvline(x<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X2D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], X2D[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
|
||||
@@ -830,7 +846,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec19">The equations </h2>
|
||||
<h2 id="___sec20">The equations </h2>
|
||||
|
||||
<p>
|
||||
Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)
|
||||
@@ -875,7 +891,7 @@ the trouble of performing the transformation
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec20">The problem to solve </h2>
|
||||
<h2 id="___sec21">The problem to solve </h2>
|
||||
Using our definition of the kernel We can rewrite again the Lagrangian
|
||||
$$
|
||||
{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j,
|
||||
@@ -911,7 +927,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec21">Different kernels and Mercer's theorem </h2>
|
||||
<h2 id="___sec22">Different kernels and Mercer's theorem </h2>
|
||||
|
||||
<p>
|
||||
There are several popular kernels being used. These are
|
||||
@@ -949,7 +965,7 @@ in practice.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec22">The moons example </h2>
|
||||
<h2 id="___sec23">The moons example </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
@@ -982,7 +998,7 @@ X, y <span style="color: #666666">=</span> make_moons(n_samples<span style="colo
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==1</span>], <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>axis(axes)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$x_2$"</span>, fontsize<span style="color: #666666">=20</span>, rotation<span style="color: #666666">=0</span>)
|
||||
|
||||
@@ -1060,7 +1076,7 @@ yk <span style="color: #666666">=</span> np<span style="color: #666666">.</span>
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">121</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>scatter(x<span style="color: #666666">=</span>[<span style="color: #666666">-2</span>, <span style="color: #666666">1</span>], y<span style="color: #666666">=</span>[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>], s<span style="color: #666666">=150</span>, alpha<span style="color: #666666">=0.5</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">"red"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==0</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">4</span>), <span style="color: #BA2121">"bs"</span>)
|
||||
@@ -1070,7 +1086,7 @@ plt<span style="color: #666666">.</span>plot(x1s, x3s, <span style="color: #BA21
|
||||
plt<span style="color: #666666">.</span>gca()<span style="color: #666666">.</span>get_yaxis()<span style="color: #666666">.</span>set_ticks([<span style="color: #666666">0</span>, <span style="color: #666666">0.25</span>, <span style="color: #666666">0.5</span>, <span style="color: #666666">0.75</span>, <span style="color: #666666">1</span>])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"Similarity"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>annotate(<span style="color: #BA2121">r'$\mathbf{x}$'</span>,
|
||||
plt<span style="color: #666666">.</span>annotate(<span style="color: #BA2121">r'$\mathbf</span><span style="color: #BB6688; font-weight: bold">{x}</span><span style="color: #BA2121">$'</span>,
|
||||
xy<span style="color: #666666">=</span>(X1D[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>], <span style="color: #666666">0</span>),
|
||||
xytext<span style="color: #666666">=</span>(<span style="color: #666666">-0.5</span>, <span style="color: #666666">0.20</span>),
|
||||
ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>,
|
||||
@@ -1082,14 +1098,14 @@ plt<span style="color: #666666">.</span>text(<span style="color: #666666">1</spa
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">122</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>axvline(x<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(XK[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==0</span>], XK[:, <span style="color: #666666">1</span>][yk<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(XK[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==1</span>], XK[:, <span style="color: #666666">1</span>][yk<span style="color: #666666">==1</span>], <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_2$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$x_3$ "</span>, fontsize<span style="color: #666666">=20</span>, rotation<span style="color: #666666">=0</span>)
|
||||
plt<span style="color: #666666">.</span>annotate(<span style="color: #BA2121">r'$\phi\left(\mathbf{x}\right)$'</span>,
|
||||
plt<span style="color: #666666">.</span>annotate(<span style="color: #BA2121">r'$\phi\left(\mathbf</span><span style="color: #BB6688; font-weight: bold">{x}</span><span style="color: #BA2121">\right)$'</span>,
|
||||
xy<span style="color: #666666">=</span>(XK[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>], XK[<span style="color: #666666">3</span>, <span style="color: #666666">1</span>]),
|
||||
xytext<span style="color: #666666">=</span>(<span style="color: #666666">0.65</span>, <span style="color: #666666">0.50</span>),
|
||||
ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>,
|
||||
@@ -1107,7 +1123,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
x1_example <span style="color: #666666">=</span> X1D[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>]
|
||||
<span style="color: #008000; font-weight: bold">for</span> landmark <span style="color: #AA22FF; font-weight: bold">in</span> (<span style="color: #666666">-2</span>, <span style="color: #666666">1</span>):
|
||||
k <span style="color: #666666">=</span> gaussian_rbf(np<span style="color: #666666">.</span>array([[x1_example]]), np<span style="color: #666666">.</span>array([[landmark]]), gamma)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Phi({}, {}) = {}"</span><span style="color: #666666">.</span>format(x1_example, landmark, k))
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Phi(</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">, </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">) = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(x1_example, landmark, k))
|
||||
|
||||
rbf_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
@@ -1138,14 +1154,14 @@ plt<span style="color: #666666">.</span>figure(figsize<span style="color: #66666
|
||||
plot_predictions(svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
gamma, C <span style="color: #666666">=</span> hyperparams[i]
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"$\gamma = {}, C = {}$"</span><span style="color: #666666">.</span>format(gamma, C), fontsize<span style="color: #666666">=16</span>)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"$\gamma = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">, C = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">$"</span><span style="color: #666666">.</span>format(gamma, C), fontsize<span style="color: #666666">=16</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec23">Mathematical optimization of convex functions </h2>
|
||||
<h2 id="___sec24">Mathematical optimization of convex functions </h2>
|
||||
|
||||
<p>
|
||||
A mathematical (quadratic) optimization problem, or just optimization problem, has the form
|
||||
@@ -1170,7 +1186,7 @@ Convex optimization problems play a central role in applied mathematics and we r
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec24">How do we solve these problems? </h2>
|
||||
<h2 id="___sec25">How do we solve these problems? </h2>
|
||||
|
||||
<p>
|
||||
If we use Python as programming language and wish to venture beyond
|
||||
@@ -1196,7 +1212,7 @@ This will make our life much easier. You don't need t write your own optimizer.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec25">A simple example </h2>
|
||||
<h2 id="___sec26">A simple example </h2>
|
||||
|
||||
<p>
|
||||
We remind ourselves about the general problem we want to solve
|
||||
@@ -1264,7 +1280,7 @@ sol[’primal objective’]
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec26">Back to the more realistic cases </h2>
|
||||
<h2 id="___sec27">Back to the more realistic cases </h2>
|
||||
|
||||
<p>
|
||||
We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the <b>slack</b> parameter \( C \) we have
|
||||
|
||||
Reference in New Issue
Block a user