Morten Hjorth-Jensen
341 lines
21 KiB
HTML
341 lines
21 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. Start Principal Component Analysis">
|
|
<title>Week 46: Support Vector Machines and Project 3. Start Principal Component Analysis</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'),
|
|
('Basic ideas of the Principal Component Analysis (PCA)',
|
|
2,
|
|
None,
|
|
'basic-ideas-of-the-principal-component-analysis-pca'),
|
|
('Introducing the Covariance and Correlation functions',
|
|
2,
|
|
None,
|
|
'introducing-the-covariance-and-correlation-functions'),
|
|
('More on the covariance', 2, None, 'more-on-the-covariance'),
|
|
('Reminding ourselves about Linear Regression',
|
|
2,
|
|
None,
|
|
'reminding-ourselves-about-linear-regression'),
|
|
('Simple Example', 2, None, 'simple-example'),
|
|
('The Correlation Matrix', 2, None, 'the-correlation-matrix'),
|
|
('Numpy Functionality', 2, None, 'numpy-functionality'),
|
|
('Correlation Matrix again', 2, None, 'correlation-matrix-again'),
|
|
('Using Pandas', 2, None, 'using-pandas'),
|
|
('And then the Franke Function',
|
|
2,
|
|
None,
|
|
'and-then-the-franke-function'),
|
|
('Links with the Design Matrix',
|
|
2,
|
|
None,
|
|
'links-with-the-design-matrix'),
|
|
('Computing the Expectation Values',
|
|
2,
|
|
None,
|
|
'computing-the-expectation-values'),
|
|
('Towards the PCA theorem', 2, None, 'towards-the-pca-theorem'),
|
|
('More on the PCA Theorem', 2, None, 'more-on-the-pca-theorem'),
|
|
('Writing our own PCA code', 2, None, 'writing-our-own-pca-code'),
|
|
('Implementing it', 2, None, 'implementing-it'),
|
|
('First Step', 2, None, 'first-step'),
|
|
('Scaling', 2, None, 'scaling'),
|
|
('Centered Data', 2, None, 'centered-data'),
|
|
('Exploring', 2, None, 'exploring'),
|
|
('Diagonalize the sample covariance matrix to obtain the '
|
|
'principal components',
|
|
2,
|
|
None,
|
|
'diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components'),
|
|
('Collecting all Steps', 2, None, 'collecting-all-steps'),
|
|
('Classical PCA Theorem', 2, None, 'classical-pca-theorem'),
|
|
('The PCA Theorem', 2, None, 'the-pca-theorem'),
|
|
('Geometric Interpretation and link with Singular Value '
|
|
'Decomposition',
|
|
2,
|
|
None,
|
|
'geometric-interpretation-and-link-with-singular-value-decomposition'),
|
|
('PCA and scikit-learn', 2, None, 'pca-and-scikit-learn'),
|
|
('Back to the Cancer Data', 2, None, 'back-to-the-cancer-data'),
|
|
('Incremental PCA', 2, None, 'incremental-pca'),
|
|
('Randomized PCA', 3, None, 'randomized-pca'),
|
|
('Kernel PCA', 3, None, 'kernel-pca'),
|
|
('Other techniques', 2, None, 'other-techniques')]}
|
|
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. Start Principal Component Analysis</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%;"><b>Overview of week 46</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs002.html#support-vector-machines-overarching-aims" style="font-size: 80%;"><b>Support Vector Machines, overarching aims</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs003.html#hyperplanes-and-all-that" style="font-size: 80%;"><b>Hyperplanes and all that</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs004.html#what-is-a-hyperplane" style="font-size: 80%;"><b>What is a hyperplane?</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs005.html#a-p-dimensional-space-of-features" style="font-size: 80%;"><b>A \( p \)-dimensional space of features</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs006.html#the-two-dimensional-case" style="font-size: 80%;"><b>The two-dimensional case</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs007.html#getting-into-the-details" style="font-size: 80%;"><b>Getting into the details</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs008.html#first-attempt-at-a-minimization-approach" style="font-size: 80%;"><b>First attempt at a minimization approach</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs009.html#solving-the-equations" style="font-size: 80%;"><b>Solving the equations</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs010.html#code-example" style="font-size: 80%;"><b>Code Example</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs011.html#problems-with-the-simpler-approach" style="font-size: 80%;"><b>Problems with the Simpler Approach</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs012.html#a-better-approach" style="font-size: 80%;"><b>A better approach</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs013.html#a-quick-reminder-on-lagrangian-multipliers" style="font-size: 80%;"><b>A quick Reminder on Lagrangian Multipliers</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs014.html#adding-the-multiplier" style="font-size: 80%;"><b>Adding the Multiplier</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs015.html#setting-up-the-problem" style="font-size: 80%;"><b>Setting up the Problem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs022.html#the-problem-to-solve" style="font-size: 80%;"><b>The problem to solve</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs017.html#the-last-steps" style="font-size: 80%;"><b>The last steps</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs018.html#a-soft-classifier" style="font-size: 80%;"><b>A soft classifier</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs019.html#soft-optmization-problem" style="font-size: 80%;"><b>Soft optmization problem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs020.html#kernels-and-non-linearity" style="font-size: 80%;"><b>Kernels and non-linearity</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs021.html#the-equations" style="font-size: 80%;"><b>The equations</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs022.html#the-problem-to-solve" style="font-size: 80%;"><b>The problem to solve</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs023.html#different-kernels-and-mercer-s-theorem" style="font-size: 80%;"><b>Different kernels and Mercer's theorem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs024.html#the-moons-example" style="font-size: 80%;"><b>The moons example</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs025.html#mathematical-optimization-of-convex-functions" style="font-size: 80%;"><b>Mathematical optimization of convex functions</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="#how-do-we-solve-these-problems" style="font-size: 80%;"><b>How do we solve these problems?</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs027.html#a-simple-example" style="font-size: 80%;"><b>A simple example</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs028.html#back-to-the-more-realistic-cases" style="font-size: 80%;"><b>Back to the more realistic cases</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs029.html#basic-ideas-of-the-principal-component-analysis-pca" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs030.html#introducing-the-covariance-and-correlation-functions" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs031.html#more-on-the-covariance" style="font-size: 80%;"><b>More on the covariance</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs032.html#reminding-ourselves-about-linear-regression" style="font-size: 80%;"><b>Reminding ourselves about Linear Regression</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs033.html#simple-example" style="font-size: 80%;"><b>Simple Example</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs034.html#the-correlation-matrix" style="font-size: 80%;"><b>The Correlation Matrix</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs035.html#numpy-functionality" style="font-size: 80%;"><b>Numpy Functionality</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs036.html#correlation-matrix-again" style="font-size: 80%;"><b>Correlation Matrix again</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs037.html#using-pandas" style="font-size: 80%;"><b>Using Pandas</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs038.html#and-then-the-franke-function" style="font-size: 80%;"><b>And then the Franke Function</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs039.html#links-with-the-design-matrix" style="font-size: 80%;"><b>Links with the Design Matrix</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs040.html#computing-the-expectation-values" style="font-size: 80%;"><b>Computing the Expectation Values</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs041.html#towards-the-pca-theorem" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs042.html#more-on-the-pca-theorem" style="font-size: 80%;"><b>More on the PCA Theorem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs043.html#writing-our-own-pca-code" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs044.html#implementing-it" style="font-size: 80%;"><b>Implementing it</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs045.html#first-step" style="font-size: 80%;"><b>First Step</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs046.html#scaling" style="font-size: 80%;"><b>Scaling</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs047.html#centered-data" style="font-size: 80%;"><b>Centered Data</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs048.html#exploring" style="font-size: 80%;"><b>Exploring</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs049.html#diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components" style="font-size: 80%;"><b>Diagonalize the sample covariance matrix to obtain the principal components</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs050.html#collecting-all-steps" style="font-size: 80%;"><b>Collecting all Steps</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs051.html#classical-pca-theorem" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs052.html#the-pca-theorem" style="font-size: 80%;"><b>The PCA Theorem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs053.html#geometric-interpretation-and-link-with-singular-value-decomposition" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs054.html#pca-and-scikit-learn" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs055.html#back-to-the-cancer-data" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs056.html#incremental-pca" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs056.html#randomized-pca" style="font-size: 80%;"> Randomized PCA</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs056.html#kernel-pca" style="font-size: 80%;"> Kernel PCA</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week46-bs057.html#other-techniques" style="font-size: 80%;"><b>Other techniques</b></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="part0026"></a>
|
|
<!-- !split -->
|
|
<h2 id="how-do-we-solve-these-problems" class="anchor">How do we solve these problems? </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.
|
|
</p>
|
|
|
|
<p>The functions we need are contained in the <a href="https://cvxopt.org/userguide/coneprog.html" target="_self">quadratic programming library CVXOPT</a> and we need to import it together with <b>numpy</b> as</p>
|
|
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="cell border-box-sizing code_cell rendered">
|
|
<div class="input">
|
|
<div class="inner_cell">
|
|
<div class="input_area">
|
|
<div class="highlight" style="background: #f8f8f8">
|
|
<pre style="line-height: 125%;"><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>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div class="output_wrapper">
|
|
<div class="output">
|
|
<div class="output_area">
|
|
<div class="output_subarea output_stream output_stdout output_text">
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<p>This will make our life much easier. You don't need t0 write your own optimizer.</p>
|
|
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._week46-bs025.html">«</a></li>
|
|
<li><a href="._week46-bs000.html">1</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._week46-bs018.html">19</a></li>
|
|
<li><a href="._week46-bs019.html">20</a></li>
|
|
<li><a href="._week46-bs020.html">21</a></li>
|
|
<li><a href="._week46-bs021.html">22</a></li>
|
|
<li><a href="._week46-bs022.html">23</a></li>
|
|
<li><a href="._week46-bs023.html">24</a></li>
|
|
<li><a href="._week46-bs024.html">25</a></li>
|
|
<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-bs029.html">30</a></li>
|
|
<li><a href="._week46-bs030.html">31</a></li>
|
|
<li><a href="._week46-bs031.html">32</a></li>
|
|
<li><a href="._week46-bs032.html">33</a></li>
|
|
<li><a href="._week46-bs033.html">34</a></li>
|
|
<li><a href="._week46-bs034.html">35</a></li>
|
|
<li><a href="._week46-bs035.html">36</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._week46-bs057.html">58</a></li>
|
|
<li><a href="._week46-bs027.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>
|
|
|