Morten Hjorth-Jensen
232 lines
12 KiB
HTML
232 lines
12 KiB
HTML
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<title>Week 46: Support Vector Machines and Project 3.</title>
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('Support Vector Machines, overarching aims',
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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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<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#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-bs003.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-bs004.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-bs005.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="._week46-bs006.html#the-two-dimensional-case" style="font-size: 80%;">The two-dimensional case</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs007.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-bs008.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-bs009.html#solving-the-equations" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs010.html#code-example" style="font-size: 80%;">Code Example</a></li>
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<!-- navigation toc: --> <li><a href="#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-bs012.html#a-better-approach" style="font-size: 80%;">A better approach</a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#adding-the-multiplier" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.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-bs022.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-bs017.html#the-last-steps" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#a-soft-classifier" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#soft-optmization-problem" style="font-size: 80%;">Soft optmization problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.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-bs021.html#the-equations" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.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-bs023.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-bs024.html#the-moons-example" style="font-size: 80%;">The moons example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.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-bs026.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-bs027.html#a-simple-example" style="font-size: 80%;">A simple example</a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._week46-bs029.html#support-vector-machines-for-regression" style="font-size: 80%;">Support vector machines for regression</a></li>
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</li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0011"></a>
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<!-- !split -->
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<h2 id="problems-with-the-simpler-approach" class="anchor">Problems with the Simpler Approach </h2>
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<p>There are however problems with this approach, although it looks
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pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.
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</p>
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<p>For small
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gaps between the entries, we may also end up needing many iterations
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before the solutions converge and if the data cannot be separated
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properly into two distinct classes, we may not experience a converge
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at all.
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