Morten Hjorth-Jensen
270 lines
13 KiB
HTML
270 lines
13 KiB
HTML
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'sections': [('Overview of week 46', 2, None, 'overview-of-week-46'),
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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="._week46-bs008.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-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="#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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</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="part0015"></a>
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<!-- !split -->
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<h2 id="a-quick-reminder-on-lagrangian-multipliers" class="anchor">A quick Reminder on Lagrangian Multipliers </h2>
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<p>Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an
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extreme we have
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</p>
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$$
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df=0.
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$$
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<p>A necessary and sufficient condition is</p>
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$$
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\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0,
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$$
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<p>due to</p>
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$$
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df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz.
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$$
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<p>In many problems the variables \( x,y,z \) are often subject to constraints (such as those above for the margin)
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so that they are no longer all independent. It is possible at least in principle to use each
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constraint to eliminate one variable
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and to proceed with a new and smaller set of independent varables.
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</p>
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<p>The use of so-called Lagrangian multipliers is an alternative technique when the elimination
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of variables is incovenient or undesirable. Assume that we have an equation of constraint on
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the variables \( x,y,z \)
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</p>
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$$
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\phi(x,y,z) = 0,
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$$
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<p> resulting in</p>
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$$
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d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0.
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$$
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<p>Now we cannot set anymore</p>
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$$
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\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0,
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$$
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<p>if \( df=0 \) is wanted
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because there are now only two independent variables! Assume \( x \) and \( y \) are the independent
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variables.
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Then \( dz \) is no longer arbitrary.
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</p>
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<p>
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