update week 46
This commit is contained in:
Morten Hjorth-Jensen
@@ -37,6 +37,14 @@ doconce format html week46.do.txt --html_style=bootstrap --pygments_html_style=d
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Overview of week 46', 2, None, 'overview-of-week-46'),
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('Eventual mini-workshop on project 3, Friday November 18',
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2,
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None,
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'eventual-mini-workshop-on-project-3-friday-november-18'),
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('Workshop topics 2021 (partly online)',
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2,
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None,
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'workshop-topics-2021-partly-online'),
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('Support Vector Machines, overarching aims',
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2,
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None,
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@@ -129,33 +137,35 @@ MathJax.Hub.Config({
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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#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="._week46-bs011.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-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="#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-bs002.html#eventual-mini-workshop-on-project-3-friday-november-18" style="font-size: 80%;">Eventual mini-workshop on project 3, Friday November 18</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#workshop-topics-2021-partly-online" style="font-size: 80%;">Workshop topics 2021 (partly online)</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="._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="#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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@@ -167,46 +177,56 @@ MathJax.Hub.Config({
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0021"></a>
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<!-- !split -->
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<h2 id="the-equations" class="anchor">The equations </h2>
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<h2 id="soft-optmization-problem" class="anchor">Soft optmization problem </h2>
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<p>Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \( x_i \) and \( y_i \) as variables)</p>
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<p>This has in turn the consequences that we change our optmization problem to finding the minimum of </p>
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$$
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z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right).
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{\cal L}=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-(1-\xi_)\right]+C\sum_{i=1}^n\xi_i-\sum_{i=1}^n\gamma_i\xi_i,
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$$
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<p>With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity)</p>
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<p>subject to </p>
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$$
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{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j,
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y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i,
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$$
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<p>subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors</p>
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<p>with the requirement \( \xi_i\geq 0 \).</p>
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<p>Taking the derivatives with respect to \( b \) and \( \boldsymbol{w} \) we obtain </p>
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$$
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y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i,
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\frac{\partial {\cal L}}{\partial b} = -\sum_{i} \lambda_iy_i=0,
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$$
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<p>from which we also find \( b \).
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To compute \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we define the kernel \( K(\boldsymbol{x}_i,\boldsymbol{x}_j) \) as
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<p>and </p>
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$$
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\frac{\partial {\cal L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i,
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$$
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<p>and</p>
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$$
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\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i.
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$$
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<p>Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before</p>
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$$
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{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j,
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$$
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<p>but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and \( 0\leq\lambda_i \leq C \).
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We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads
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</p>
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$$
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K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j).
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\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i,
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$$
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<p>For the above example, the kernel reads</p>
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$$
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K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2.
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\gamma_i\xi_i = 0,
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$$
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<p>We note that this is nothing but the dot product of the two original
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vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the
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product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute
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the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \).
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</p>
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<p>and </p>
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$$
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y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i.
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$$
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<p>This leads to the so-called
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kernel trick and the result leads to the same as if we went through
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the trouble of performing the transformation
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\( \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations.
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</p>
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<p>
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<!-- navigation buttons at the bottom of the page -->
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@@ -230,6 +250,8 @@ the trouble of performing the transformation
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<li><a href="._week46-bs026.html">27</a></li>
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<li><a href="._week46-bs027.html">28</a></li>
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<li><a href="._week46-bs028.html">29</a></li>
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<li><a href="._week46-bs029.html">30</a></li>
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<li><a href="._week46-bs030.html">31</a></li>
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<li><a href="._week46-bs022.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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