Morten Hjorth-Jensen
336 lines
22 KiB
HTML
336 lines
22 KiB
HTML
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('Support Vector Machines, overarching aims',
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2,
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None,
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('A $p$-dimensional space of features',
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('First attempt at a minimization approach',
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('Adding the Multiplier', 2, None, 'adding-the-multiplier'),
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('Setting up the Problem', 2, None, 'setting-up-the-problem'),
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('The problem to solve', 2, None, 'the-problem-to-solve'),
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('The last steps', 2, None, 'the-last-steps'),
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('A soft classifier', 2, None, 'a-soft-classifier'),
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('Soft optmization problem', 2, None, 'soft-optmization-problem'),
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('Kernels and non-linearity',
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("Different kernels and Mercer's theorem",
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('The moons example', 2, None, 'the-moons-example'),
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('Mathematical optimization of convex functions',
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('A simple example', 2, None, 'a-simple-example'),
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('Back to the more realistic cases',
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'back-to-the-more-realistic-cases'),
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('Basic ideas of the Principal Component Analysis (PCA)',
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('The Correlation Matrix', 2, None, 'the-correlation-matrix'),
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('Numpy Functionality', 2, None, 'numpy-functionality'),
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('Using Pandas', 2, None, 'using-pandas'),
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('And then the Franke Function',
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2,
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('Links with the Design Matrix',
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('Computing the Expectation Values',
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('Towards the PCA theorem', 2, None, 'towards-the-pca-theorem'),
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('More on the PCA Theorem', 2, None, 'more-on-the-pca-theorem'),
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('Writing our own PCA code', 2, None, 'writing-our-own-pca-code'),
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('Implementing it', 2, None, 'implementing-it'),
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('Diagonalize the sample covariance matrix to obtain the '
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('Collecting all Steps', 2, None, 'collecting-all-steps'),
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('Classical PCA Theorem', 2, None, 'classical-pca-theorem'),
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('The PCA Theorem', 2, None, 'the-pca-theorem'),
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('Geometric Interpretation and link with Singular Value '
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('Incremental PCA', 2, None, 'incremental-pca'),
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('Randomized PCA', 3, None, 'randomized-pca'),
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('Kernel PCA', 3, None, 'kernel-pca'),
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('Other techniques', 2, None, 'other-techniques')]}
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week46-bs001.html#overview-of-week-46" style="font-size: 80%;"><b>Overview of week 46</b></a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#hyperplanes-and-all-that" style="font-size: 80%;"><b>Hyperplanes and all that</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs004.html#what-is-a-hyperplane" style="font-size: 80%;"><b>What is a hyperplane?</b></a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs006.html#the-two-dimensional-case" style="font-size: 80%;"><b>The two-dimensional case</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs007.html#getting-into-the-details" style="font-size: 80%;"><b>Getting into the details</b></a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs009.html#solving-the-equations" style="font-size: 80%;"><b>Solving the equations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs010.html#code-example" style="font-size: 80%;"><b>Code Example</b></a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs012.html#a-better-approach" style="font-size: 80%;"><b>A better approach</b></a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#adding-the-multiplier" style="font-size: 80%;"><b>Adding the Multiplier</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.html#setting-up-the-problem" style="font-size: 80%;"><b>Setting up the Problem</b></a></li>
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<!-- navigation toc: --> <li><a href="#the-problem-to-solve" style="font-size: 80%;"><b>The problem to solve</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#the-last-steps" style="font-size: 80%;"><b>The last steps</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#a-soft-classifier" style="font-size: 80%;"><b>A soft classifier</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#soft-optmization-problem" style="font-size: 80%;"><b>Soft optmization problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#kernels-and-non-linearity" style="font-size: 80%;"><b>Kernels and non-linearity</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#the-equations" style="font-size: 80%;"><b>The equations</b></a></li>
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<!-- navigation toc: --> <li><a href="#the-problem-to-solve" style="font-size: 80%;"><b>The problem to solve</b></a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#the-moons-example" style="font-size: 80%;"><b>The moons example</b></a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs026.html#how-do-we-solve-these-problems" style="font-size: 80%;"><b>How do we solve these problems?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#a-simple-example" style="font-size: 80%;"><b>A simple example</b></a></li>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs031.html#more-on-the-covariance" style="font-size: 80%;"><b>More on the covariance</b></a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs033.html#simple-example" style="font-size: 80%;"><b>Simple Example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs034.html#the-correlation-matrix" style="font-size: 80%;"><b>The Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs035.html#numpy-functionality" style="font-size: 80%;"><b>Numpy Functionality</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs036.html#correlation-matrix-again" style="font-size: 80%;"><b>Correlation Matrix again</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs037.html#using-pandas" style="font-size: 80%;"><b>Using Pandas</b></a></li>
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<!-- 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>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs040.html#computing-the-expectation-values" style="font-size: 80%;"><b>Computing the Expectation Values</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs041.html#towards-the-pca-theorem" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
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<!-- 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>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs044.html#implementing-it" style="font-size: 80%;"><b>Implementing it</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs045.html#first-step" style="font-size: 80%;"><b>First Step</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs046.html#scaling" style="font-size: 80%;"><b>Scaling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs047.html#centered-data" style="font-size: 80%;"><b>Centered Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs048.html#exploring" style="font-size: 80%;"><b>Exploring</b></a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs050.html#collecting-all-steps" style="font-size: 80%;"><b>Collecting all Steps</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs051.html#classical-pca-theorem" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs052.html#the-pca-theorem" style="font-size: 80%;"><b>The PCA Theorem</b></a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs054.html#pca-and-scikit-learn" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs056.html#incremental-pca" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs056.html#randomized-pca" style="font-size: 80%;"> Randomized PCA</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs056.html#kernel-pca" style="font-size: 80%;"> Kernel PCA</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs057.html#other-techniques" style="font-size: 80%;"><b>Other techniques</b></a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div> <!-- end of navigation bar -->
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0022"></a>
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<!-- !split -->
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<h2 id="the-problem-to-solve" class="anchor">The problem to solve </h2>
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<p>Using our definition of the kernel We can rewrite again the Lagrangian</p>
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$$
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{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j,
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$$
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<p>subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) in terms of a convex optimization problem</p>
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$$
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\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\
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y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\
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\dots & \dots & \dots & \dots & \dots \\
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\dots & \dots & \dots & \dots & \dots \\
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y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\
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\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda},
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$$
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<p>subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and
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\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \).
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If we add the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \).
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</p>
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<p>We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type</p>
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$$
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\begin{align*}
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&\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber
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&\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f.
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\end{align*}
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$$
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<p>Below we discuss how to solve these equations. Here we note that the matrix \( \boldsymbol{P} \) has matrix elements \( p_{ij}=y_iy_jK(\boldsymbol{x}_i,\boldsymbol{x}_j) \).
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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
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\( 0\leq \lambda_i \) and \( \lambda_i \leq C \). These two inequalities define then the matrix \( \boldsymbol{G} \) and the vector \( \boldsymbol{h} \).
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</p>
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