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update week 46

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Morten Hjorth-Jensen
2022-11-13 10:15:13 +01:00
parent 8ac7d9e4af
commit ff85f76c97
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@@ -37,6 +37,14 @@ doconce format html week46.do.txt --html_style=bootstrap --pygments_html_style=d
<!-- tocinfo
{'highest level': 2,
'sections': [('Overview of week 46', 2, None, 'overview-of-week-46'),
('Eventual mini-workshop on project 3, Friday November 18',
2,
None,
'eventual-mini-workshop-on-project-3-friday-november-18'),
('Workshop topics 2021 (partly online)',
2,
None,
'workshop-topics-2021-partly-online'),
('Support Vector Machines, overarching aims',
2,
None,
@@ -129,33 +137,35 @@ MathJax.Hub.Config({
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
<ul class="dropdown-menu">
<!-- navigation toc: --> <li><a href="._week46-bs001.html#overview-of-week-46" style="font-size: 80%;">Overview of week 46</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs002.html#support-vector-machines-overarching-aims" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs003.html#hyperplanes-and-all-that" style="font-size: 80%;">Hyperplanes and all that</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs004.html#what-is-a-hyperplane" style="font-size: 80%;">What is a hyperplane?</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week46-bs006.html#the-two-dimensional-case" style="font-size: 80%;">The two-dimensional case</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs007.html#getting-into-the-details" style="font-size: 80%;">Getting into the details</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week46-bs009.html#solving-the-equations" style="font-size: 80%;">Solving the equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs010.html#code-example" style="font-size: 80%;">Code Example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs011.html#problems-with-the-simpler-approach" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs012.html#a-better-approach" style="font-size: 80%;">A better approach</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week46-bs014.html#adding-the-multiplier" style="font-size: 80%;">Adding the Multiplier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs015.html#setting-up-the-problem" style="font-size: 80%;">Setting up the Problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs022.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs017.html#the-last-steps" style="font-size: 80%;">The last steps</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs018.html#a-soft-classifier" style="font-size: 80%;">A soft classifier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs019.html#soft-optmization-problem" style="font-size: 80%;">Soft optmization problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs020.html#kernels-and-non-linearity" style="font-size: 80%;">Kernels and non-linearity</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs021.html#the-equations" style="font-size: 80%;">The equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs022.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week46-bs024.html#the-moons-example" style="font-size: 80%;">The moons example</a></li>
<!-- navigation toc: --> <li><a href="#mathematical-optimization-of-convex-functions" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week46-bs027.html#a-simple-example" style="font-size: 80%;">A simple example</a></li>
<!-- 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>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week46-bs003.html#workshop-topics-2021-partly-online" style="font-size: 80%;">Workshop topics 2021 (partly online)</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs004.html#support-vector-machines-overarching-aims" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs005.html#hyperplanes-and-all-that" style="font-size: 80%;">Hyperplanes and all that</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs006.html#what-is-a-hyperplane" style="font-size: 80%;">What is a hyperplane?</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week46-bs008.html#the-two-dimensional-case" style="font-size: 80%;">The two-dimensional case</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs009.html#getting-into-the-details" style="font-size: 80%;">Getting into the details</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week46-bs011.html#solving-the-equations" style="font-size: 80%;">Solving the equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs012.html#code-example" style="font-size: 80%;">Code Example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs013.html#problems-with-the-simpler-approach" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs014.html#a-better-approach" style="font-size: 80%;">A better approach</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week46-bs016.html#adding-the-multiplier" style="font-size: 80%;">Adding the Multiplier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs017.html#setting-up-the-problem" style="font-size: 80%;">Setting up the Problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs024.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs019.html#the-last-steps" style="font-size: 80%;">The last steps</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs020.html#a-soft-classifier" style="font-size: 80%;">A soft classifier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs021.html#soft-optmization-problem" style="font-size: 80%;">Soft optmization problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs022.html#kernels-and-non-linearity" style="font-size: 80%;">Kernels and non-linearity</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs023.html#the-equations" style="font-size: 80%;">The equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs024.html#the-problem-to-solve" style="font-size: 80%;">The problem to solve</a></li>
<!-- navigation toc: --> <li><a href="#different-kernels-and-mercer-s-theorem" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs026.html#the-moons-example" style="font-size: 80%;">The moons example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs027.html#mathematical-optimization-of-convex-functions" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._week46-bs029.html#a-simple-example" style="font-size: 80%;">A simple example</a></li>
<!-- 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>
</ul>
</li>
@@ -167,26 +177,37 @@ MathJax.Hub.Config({
<p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <!-- add vertical space -->
<a name="part0025"></a>
<!-- !split -->
<h2 id="mathematical-optimization-of-convex-functions" class="anchor">Mathematical optimization of convex functions </h2>
<h2 id="different-kernels-and-mercer-s-theorem" class="anchor">Different kernels and Mercer's theorem </h2>
<p>A mathematical (quadratic) optimization problem, or just optimization problem, has the form</p>
$$
\begin{align*}
&\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber
&\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f.
\end{align*}
$$
<p>There are several popular kernels being used. These are</p>
<ol>
<li> Linear: \( K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y} \),</li>
<li> Polynomial: \( K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d \),</li>
<li> Gaussian Radial Basis Function: \( K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)} \),</li>
<li> Tanh: \( K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)} \),</li>
</ol>
<p>and many other ones.</p>
<p>subject to some constraints for say a selected set \( i=1,2,\dots, n \).
In our case we are optimizing with respect to the Lagrangian multipliers \( \lambda_i \), and the
vector \( \boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n] \) is the optimization variable we are dealing with.
<p>An important theorem for us is <a href="https://en.wikipedia.org/wiki/Mercer%27s_theorem" target="_self">Mercer's
theorem</a>. The
theorem states that if a kernel function \( K \) is symmetric, continuous
and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then there
exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into
another space (possibly with much higher dimensions) such that
</p>
<p>In our case we are particularly interested in a class of optimization problems called convex optmization problems.
In our discussion on gradient descent methods we discussed at length the definition of a convex function.
$$
K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j).
$$
<p>So you can use \( K \) as a kernel since you know \( \phi \) exists, even if
you don&#8217;t know what \( \phi \) is.
</p>
<p>Convex optimization problems play a central role in applied mathematics and we recommend strongly <a href="http://web.stanford.edu/~boyd/cvxbook/" target="_self">Boyd and Vandenberghe's text on the topics</a>.</p>
<p>Note that some frequently used kernels (such as the Sigmoid kernel)
don&#8217;t respect all of Mercer&#8217;s conditions, yet they generally work well
in practice.
</p>
<p>
<!-- navigation buttons at the bottom of the page -->
@@ -206,6 +227,8 @@ In our discussion on gradient descent methods we discussed at length the definit
<li><a href="._week46-bs026.html">27</a></li>
<li><a href="._week46-bs027.html">28</a></li>
<li><a href="._week46-bs028.html">29</a></li>
<li><a href="._week46-bs029.html">30</a></li>
<li><a href="._week46-bs030.html">31</a></li>
<li><a href="._week46-bs026.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->