updating week 46
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mhjensen
@@ -42,39 +42,41 @@ Automatically generated HTML file from DocOnce source
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Overview of week 46', 2, None, '___sec0'),
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('Support Vector Machines, overarching aims', 2, None, '___sec1'),
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('Hyperplanes and all that', 2, None, '___sec2'),
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('What is a hyperplane?', 2, None, '___sec3'),
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('A $p$-dimensional space of features', 2, None, '___sec4'),
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('The two-dimensional case', 2, None, '___sec5'),
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('Getting into the details', 2, None, '___sec6'),
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('First attempt at a minimization approach', 2, None, '___sec7'),
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('Solving the equations', 2, None, '___sec8'),
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('Code Example', 2, None, '___sec9'),
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('Problems with the Simpler Approach', 2, None, '___sec10'),
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('A better approach', 2, None, '___sec11'),
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('Thursday', 2, None, '___sec1'),
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('Friday', 2, None, '___sec2'),
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('Support Vector Machines, overarching aims', 2, None, '___sec3'),
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('Hyperplanes and all that', 2, None, '___sec4'),
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('What is a hyperplane?', 2, None, '___sec5'),
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('A $p$-dimensional space of features', 2, None, '___sec6'),
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('The two-dimensional case', 2, None, '___sec7'),
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('Getting into the details', 2, None, '___sec8'),
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('First attempt at a minimization approach', 2, None, '___sec9'),
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('Solving the equations', 2, None, '___sec10'),
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('Code Example', 2, None, '___sec11'),
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('Problems with the Simpler Approach', 2, None, '___sec12'),
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('A better approach', 2, None, '___sec13'),
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('A quick Reminder on Lagrangian Multipliers',
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2,
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None,
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'___sec12'),
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('Adding the Multiplier', 2, None, '___sec13'),
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('Setting up the Problem', 2, None, '___sec14'),
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('The problem to solve', 2, None, '___sec15'),
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('The last steps', 2, None, '___sec16'),
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('A soft classifier', 2, None, '___sec17'),
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('Soft optmization problem', 2, None, '___sec18'),
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('Kernels and non-linearity', 2, None, '___sec19'),
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('The equations', 2, None, '___sec20'),
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('The problem to solve', 2, None, '___sec21'),
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("Different kernels and Mercer's theorem", 2, None, '___sec22'),
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('The moons example', 2, None, '___sec23'),
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'___sec14'),
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('Adding the Multiplier', 2, None, '___sec15'),
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('Setting up the Problem', 2, None, '___sec16'),
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('The problem to solve', 2, None, '___sec17'),
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('The last steps', 2, None, '___sec18'),
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('A soft classifier', 2, None, '___sec19'),
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('Soft optmization problem', 2, None, '___sec20'),
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('Kernels and non-linearity', 2, None, '___sec21'),
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('The equations', 2, None, '___sec22'),
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('The problem to solve', 2, None, '___sec23'),
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("Different kernels and Mercer's theorem", 2, None, '___sec24'),
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('The moons example', 2, None, '___sec25'),
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('Mathematical optimization of convex functions',
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2,
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None,
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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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'___sec26'),
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('How do we solve these problems?', 2, None, '___sec27'),
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('A simple example', 2, None, '___sec28'),
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('Back to the more realistic cases', 2, None, '___sec29')]}
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end of tocinfo -->
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<body>
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@@ -113,33 +115,35 @@ MathJax.Hub.Config({
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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">
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<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
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<!-- 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="._week46-bs010.html#___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="#___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>
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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-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>
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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>
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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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<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Thursday</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Friday</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">Hyperplanes and all that</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">What is a hyperplane?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">The two-dimensional case</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Getting into the details</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">A better approach</a></li>
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<!-- navigation toc: --> <li><a href="#___sec14" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">Setting up the Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">Soft optmization problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Kernels and non-linearity</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">The moons example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" 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%;">How do we solve these problems?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs029.html#___sec28" style="font-size: 80%;">A simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs030.html#___sec29" 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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@@ -155,43 +159,52 @@ MathJax.Hub.Config({
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<a name="part0015"></a>
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<!-- !split -->
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<h2 id="___sec14" class="anchor">Setting up the Problem </h2>
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In order to solve the above problem, we define the following Lagrangian function to be minimized
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$$
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{\cal L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right],
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$$
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where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \).
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<h2 id="___sec14" class="anchor">A quick Reminder on Lagrangian Multipliers </h2>
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<p>
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Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain
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Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
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extreme we have
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$$
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\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0,
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df=0.
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$$
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and
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A necessary and sufficient condition is
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$$
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\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i.
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\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0,
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$$
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Inserting these constraints into the equation for \( {\cal L} \) we obtain
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due to
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$$
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{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j,
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df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz.
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$$
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subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \).
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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
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In many problems the variables \( x,y,z \) are often subject to constraints (such as those above for the margin)
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so that they are no longer all independent. It is possible at least in principle to use each
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constraint to eliminate one variable
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and to proceed with a new and smaller set of independent varables.
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<p>
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The use of so-called Lagrangian multipliers is an alternative technique when the elimination
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of variables is incovenient or undesirable. Assume that we have an equation of constraint on
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the variables \( x,y,z \)
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$$
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\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -1\right] \hspace{0.1cm}\forall i.
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\phi(x,y,z) = 0,
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$$
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resulting in
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$$
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d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0.
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$$
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<ol>
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<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>
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<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>
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</ol>
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Now we cannot set anymore
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$$
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\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0,
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$$
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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 \( M \).
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if \( df=0 \) is wanted
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because there are now only two independent variables! Assume \( x \) and \( y \) are the independent
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variables.
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Then \( dz \) is no longer arbitrary.
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<p>
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<p>
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@@ -219,7 +232,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
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<li><a href="._week46-bs023.html">24</a></li>
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<li><a href="._week46-bs024.html">25</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week46-bs028.html">29</a></li>
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<li><a href="._week46-bs030.html">31</a></li>
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<li><a href="._week46-bs016.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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