Morten Hjorth-Jensen
241 lines
13 KiB
HTML
241 lines
13 KiB
HTML
<!--
|
|
HTML file automatically generated from DocOnce source
|
|
(https://github.com/doconce/doconce/)
|
|
doconce format html week46.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week46-bs --no_mako
|
|
-->
|
|
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
|
<meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
|
<meta name="description" content="Week 46: Support Vector Machines and Project 3.">
|
|
<title>Week 46: Support Vector Machines and Project 3.</title>
|
|
<!-- Bootstrap style: bootstrap -->
|
|
<!-- doconce format html week46.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week46-bs --no_mako -->
|
|
<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': [('Overview of week 46', 2, None, 'overview-of-week-46'),
|
|
('Support Vector Machines, overarching aims',
|
|
2,
|
|
None,
|
|
'support-vector-machines-overarching-aims'),
|
|
('Hyperplanes and all that', 2, None, 'hyperplanes-and-all-that'),
|
|
('What is a hyperplane?', 2, None, 'what-is-a-hyperplane'),
|
|
('A $p$-dimensional space of features',
|
|
2,
|
|
None,
|
|
'a-p-dimensional-space-of-features'),
|
|
('The two-dimensional case', 2, None, 'the-two-dimensional-case'),
|
|
('Getting into the details', 2, None, 'getting-into-the-details'),
|
|
('First attempt at a minimization approach',
|
|
2,
|
|
None,
|
|
'first-attempt-at-a-minimization-approach'),
|
|
('Solving the equations', 2, None, 'solving-the-equations'),
|
|
('Code Example', 2, None, 'code-example'),
|
|
('Problems with the Simpler Approach',
|
|
2,
|
|
None,
|
|
'problems-with-the-simpler-approach'),
|
|
('A better approach', 2, None, 'a-better-approach'),
|
|
('A quick Reminder on Lagrangian Multipliers',
|
|
2,
|
|
None,
|
|
'a-quick-reminder-on-lagrangian-multipliers'),
|
|
('Adding the Multiplier', 2, None, 'adding-the-multiplier'),
|
|
('Setting up the Problem', 2, None, 'setting-up-the-problem'),
|
|
('The problem to solve', 2, None, 'the-problem-to-solve'),
|
|
('The last steps', 2, None, 'the-last-steps'),
|
|
('A soft classifier', 2, None, 'a-soft-classifier'),
|
|
('Soft optmization problem', 2, None, 'soft-optmization-problem'),
|
|
('Kernels and non-linearity',
|
|
2,
|
|
None,
|
|
'kernels-and-non-linearity'),
|
|
('The equations', 2, None, 'the-equations'),
|
|
('The problem to solve', 2, None, 'the-problem-to-solve'),
|
|
("Different kernels and Mercer's theorem",
|
|
2,
|
|
None,
|
|
'different-kernels-and-mercer-s-theorem'),
|
|
('The moons example', 2, None, 'the-moons-example'),
|
|
('Mathematical optimization of convex functions',
|
|
2,
|
|
None,
|
|
'mathematical-optimization-of-convex-functions'),
|
|
('How do we solve these problems?',
|
|
2,
|
|
None,
|
|
'how-do-we-solve-these-problems'),
|
|
('A simple example', 2, None, 'a-simple-example'),
|
|
('Back to the more realistic cases',
|
|
2,
|
|
None,
|
|
'back-to-the-more-realistic-cases'),
|
|
('Support vector machines for regression',
|
|
2,
|
|
None,
|
|
'support-vector-machines-for-regression')]}
|
|
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 and Project 3.</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#overview-of-week-46" style="font-size: 80%;">Overview of week 46</a></li>
|
|
<!-- navigation toc: --> <li><a href="#support-vector-machines-overarching-aims" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs003.html#hyperplanes-and-all-that" style="font-size: 80%;">Hyperplanes and all that</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs004.html#what-is-a-hyperplane" style="font-size: 80%;">What is a hyperplane?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs005.html#a-p-dimensional-space-of-features" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs006.html#the-two-dimensional-case" style="font-size: 80%;">The two-dimensional case</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs007.html#getting-into-the-details" style="font-size: 80%;">Getting into the details</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs008.html#first-attempt-at-a-minimization-approach" style="font-size: 80%;">First attempt at a minimization approach</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs009.html#solving-the-equations" style="font-size: 80%;">Solving the equations</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs010.html#code-example" style="font-size: 80%;">Code Example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs011.html#problems-with-the-simpler-approach" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs012.html#a-better-approach" style="font-size: 80%;">A better approach</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs013.html#a-quick-reminder-on-lagrangian-multipliers" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs014.html#adding-the-multiplier" style="font-size: 80%;">Adding the Multiplier</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs015.html#setting-up-the-problem" style="font-size: 80%;">Setting up the Problem</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs022.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs017.html#the-last-steps" style="font-size: 80%;">The last steps</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs018.html#a-soft-classifier" style="font-size: 80%;">A soft classifier</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs019.html#soft-optmization-problem" style="font-size: 80%;">Soft optmization problem</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs020.html#kernels-and-non-linearity" style="font-size: 80%;">Kernels and non-linearity</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs021.html#the-equations" style="font-size: 80%;">The equations</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs022.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs023.html#different-kernels-and-mercer-s-theorem" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs024.html#the-moons-example" style="font-size: 80%;">The moons example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs025.html#mathematical-optimization-of-convex-functions" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs026.html#how-do-we-solve-these-problems" style="font-size: 80%;">How do we solve these problems?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs027.html#a-simple-example" style="font-size: 80%;">A simple example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs028.html#back-to-the-more-realistic-cases" style="font-size: 80%;">Back to the more realistic cases</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs029.html#support-vector-machines-for-regression" style="font-size: 80%;">Support vector machines for regression</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="part0002"></a>
|
|
<!-- !split -->
|
|
<h2 id="support-vector-machines-overarching-aims" class="anchor">Support Vector Machines, overarching aims </h2>
|
|
|
|
<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>
|
|
|
|
<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>
|
|
|
|
<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>
|
|
|
|
<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.
|
|
</p>
|
|
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._week46-bs001.html">«</a></li>
|
|
<li><a href="._week46-bs000.html">1</a></li>
|
|
<li><a href="._week46-bs001.html">2</a></li>
|
|
<li class="active"><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><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="">...</a></li>
|
|
<li><a href="._week46-bs029.html">30</a></li>
|
|
<li><a href="._week46-bs003.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="https://..."><img width="250" align=right src="https://..."></a>
|
|
</footer>
|
|
-->
|
|
<center style="font-size:80%">
|
|
<!-- copyright only on the titlepage -->
|
|
</center>
|
|
</body>
|
|
</html>
|
|
|