Morten Hjorth-Jensen
243 lines
13 KiB
HTML
243 lines
13 KiB
HTML
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<title>Week 46: Support Vector Machines and Project 3</title>
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines and Project 3</a>
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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#overview-of-week-46" style="font-size: 80%;">Overview of week 46</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs002.html#friday" style="font-size: 80%;">Friday</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#workshop-plan-friday-november-19-and-the-rest-of-the-lecture" style="font-size: 80%;">Workshop plan Friday November 19 and the rest of the lecture</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs004.html#support-vector-machines-overarching-aims" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs005.html#hyperplanes-and-all-that" style="font-size: 80%;">Hyperplanes and all that</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs006.html#what-is-a-hyperplane" style="font-size: 80%;">What is a hyperplane?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs007.html#a-p-dimensional-space-of-features" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
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<!-- navigation toc: --> <li><a href="#the-two-dimensional-case" style="font-size: 80%;">The two-dimensional case</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs009.html#getting-into-the-details" style="font-size: 80%;">Getting into the details</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs010.html#first-attempt-at-a-minimization-approach" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs011.html#solving-the-equations" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs012.html#code-example" style="font-size: 80%;">Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs013.html#problems-with-the-simpler-approach" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#a-better-approach" style="font-size: 80%;">A better approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.html#a-quick-reminder-on-lagrangian-multipliers" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs016.html#adding-the-multiplier" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#setting-up-the-problem" style="font-size: 80%;">Setting up the Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#the-last-steps" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#a-soft-classifier" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#soft-optmization-problem" style="font-size: 80%;">Soft optmization problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.html#kernels-and-non-linearity" style="font-size: 80%;">Kernels and non-linearity</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#the-equations" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#different-kernels-and-mercer-s-theorem" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs026.html#the-moons-example" style="font-size: 80%;">The moons example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#mathematical-optimization-of-convex-functions" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs028.html#how-do-we-solve-these-problems" style="font-size: 80%;">How do we solve these problems?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs029.html#a-simple-example" style="font-size: 80%;">A simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs030.html#back-to-the-more-realistic-cases" 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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</ul>
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</div>
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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="part0008"></a>
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<!-- !split -->
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<h2 id="the-two-dimensional-case" class="anchor">The two-dimensional case </h2>
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<p>Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional
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plane. To separate the two classes of data points, there are many
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possible lines (hyperplanes if you prefer a more strict naming)
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that could be chosen. Our objective is to find a
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plane that has the maximum margin, i.e the maximum distance between
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data points of both classes. Maximizing the margin distance provides
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some reinforcement so that future data points can be classified with
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more confidence.
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</p>
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<p>What a linear classifier attempts to accomplish is to split the
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feature space into two half spaces by placing a hyperplane between the
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data points. This hyperplane will be our decision boundary. All
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points on one side of the plane will belong to class one and all points
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on the other side of the plane will belong to the second class two.
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</p>
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<p>Unfortunately there are many ways in which we can place a hyperplane
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to divide the data. Below is an example of two candidate hyperplanes
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for our data sample.
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</p>
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<p>
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<li><a href="._week46-bs007.html">«</a></li>
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<li><a href="._week46-bs000.html">1</a></li>
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<li><a href="._week46-bs001.html">2</a></li>
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<li><a href="._week46-bs002.html">3</a></li>
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<li><a href="._week46-bs003.html">4</a></li>
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<li><a href="._week46-bs004.html">5</a></li>
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<li><a href="._week46-bs005.html">6</a></li>
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<li><a href="._week46-bs006.html">7</a></li>
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<li><a href="._week46-bs007.html">8</a></li>
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<li class="active"><a href="._week46-bs008.html">9</a></li>
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<li><a href="._week46-bs009.html">10</a></li>
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<li><a href="._week46-bs010.html">11</a></li>
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<li><a href="._week46-bs011.html">12</a></li>
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<li><a href="._week46-bs012.html">13</a></li>
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<li><a href="._week46-bs013.html">14</a></li>
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<li><a href="._week46-bs014.html">15</a></li>
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<li><a href="._week46-bs015.html">16</a></li>
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<li><a href="._week46-bs016.html">17</a></li>
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<li><a href="._week46-bs017.html">18</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week46-bs030.html">31</a></li>
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