mhjensen
223 lines
10 KiB
HTML
223 lines
10 KiB
HTML
<!--
|
|
Automatically generated HTML file from DocOnce source
|
|
(https://github.com/hplgit/doconce/)
|
|
-->
|
|
<html>
|
|
<head>
|
|
<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">
|
|
|
|
<title>Week 46: Support Vector Machines</title>
|
|
|
|
<!-- Bootstrap style: bootstrap -->
|
|
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
|
<!-- not necessary
|
|
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
|
-->
|
|
|
|
<style type="text/css">
|
|
|
|
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
|
.dropdown-menu {
|
|
height: auto;
|
|
max-height: 400px;
|
|
overflow-x: hidden;
|
|
}
|
|
|
|
/* Adds an invisible element before each target to offset for the navigation
|
|
bar */
|
|
.anchor::before {
|
|
content:"";
|
|
display:block;
|
|
height:50px; /* fixed header height for style bootstrap */
|
|
margin:-50px 0 0; /* negative fixed header height */
|
|
}
|
|
</style>
|
|
|
|
|
|
</head>
|
|
|
|
<!-- 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'),
|
|
('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'),
|
|
('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')]}
|
|
end of tocinfo -->
|
|
|
|
<body>
|
|
|
|
|
|
|
|
<script type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
TeX: {
|
|
equationNumbers: { autoNumber: "none" },
|
|
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
|
}
|
|
});
|
|
</script>
|
|
<script type="text/javascript" async
|
|
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
|
</script>
|
|
|
|
|
|
|
|
|
|
<!-- Bootstrap navigation bar -->
|
|
<div class="navbar navbar-default navbar-fixed-top">
|
|
<div class="navbar-header">
|
|
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
|
<span class="icon-bar"></span>
|
|
<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>
|
|
</div>
|
|
|
|
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
|
<ul class="nav navbar-nav navbar-right">
|
|
<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="#___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>
|
|
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
</div> <!-- end of navigation bar -->
|
|
|
|
<div class="container">
|
|
|
|
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
|
|
|
<a name="part0006"></a>
|
|
<!-- !split -->
|
|
|
|
<h2 id="___sec5" class="anchor">Getting into the details </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.
|
|
|
|
<p>
|
|
Any point defined by \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_2 \) on the line \( L \) will satisfy \( \boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0 \).
|
|
|
|
<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
|
|
$$
|
|
\delta = \frac{1}{\vert\vert \boldsymbol{w}\vert\vert}(\boldsymbol{w}^T\boldsymbol{x}+b).
|
|
$$
|
|
|
|
<p>
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._week46-bs005.html">«</a></li>
|
|
<li><a href="._week46-bs000.html">1</a></li>
|
|
<li><a href="._week46-bs001.html">2</a></li>
|
|
<li><a href="._week46-bs002.html">3</a></li>
|
|
<li><a href="._week46-bs003.html">4</a></li>
|
|
<li><a href="._week46-bs004.html">5</a></li>
|
|
<li><a href="._week46-bs005.html">6</a></li>
|
|
<li class="active"><a href="._week46-bs006.html">7</a></li>
|
|
<li><a href="._week46-bs007.html">8</a></li>
|
|
<li><a href="._week46-bs008.html">9</a></li>
|
|
<li><a href="._week46-bs009.html">10</a></li>
|
|
<li><a href="._week46-bs010.html">11</a></li>
|
|
<li><a href="._week46-bs011.html">12</a></li>
|
|
<li><a href="._week46-bs012.html">13</a></li>
|
|
<li><a href="._week46-bs013.html">14</a></li>
|
|
<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-bs007.html">»</a></li>
|
|
</ul>
|
|
<!-- ------------------- end of main content --------------- -->
|
|
|
|
</div> <!-- end container -->
|
|
<!-- include javascript, jQuery *first* -->
|
|
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
|
|
<script src="https://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
|
|
|
|
<!-- Bootstrap footer
|
|
<footer>
|
|
<a href="http://..."><img width="250" align=right src="http://..."></a>
|
|
</footer>
|
|
-->
|
|
|
|
|
|
<center style="font-size:80%">
|
|
<!-- copyright only on the titlepage -->
|
|
</center>
|
|
|
|
|
|
</body>
|
|
</html>
|
|
|
|
|