updating week 46
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mhjensen
@@ -42,39 +42,41 @@ Automatically generated HTML file from DocOnce source
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Overview of week 46', 2, None, '___sec0'),
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('Support Vector Machines, overarching aims', 2, None, '___sec1'),
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('Hyperplanes and all that', 2, None, '___sec2'),
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('What is a hyperplane?', 2, None, '___sec3'),
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('A $p$-dimensional space of features', 2, None, '___sec4'),
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('The two-dimensional case', 2, None, '___sec5'),
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('Getting into the details', 2, None, '___sec6'),
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('First attempt at a minimization approach', 2, None, '___sec7'),
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('Solving the equations', 2, None, '___sec8'),
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('Code Example', 2, None, '___sec9'),
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('Problems with the Simpler Approach', 2, None, '___sec10'),
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('A better approach', 2, None, '___sec11'),
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('Thursday', 2, None, '___sec1'),
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('Friday', 2, None, '___sec2'),
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('Support Vector Machines, overarching aims', 2, None, '___sec3'),
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('Hyperplanes and all that', 2, None, '___sec4'),
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('What is a hyperplane?', 2, None, '___sec5'),
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('A $p$-dimensional space of features', 2, None, '___sec6'),
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('The two-dimensional case', 2, None, '___sec7'),
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('Getting into the details', 2, None, '___sec8'),
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('First attempt at a minimization approach', 2, None, '___sec9'),
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('Solving the equations', 2, None, '___sec10'),
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('Code Example', 2, None, '___sec11'),
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('Problems with the Simpler Approach', 2, None, '___sec12'),
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('A better approach', 2, None, '___sec13'),
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('A quick Reminder on Lagrangian Multipliers',
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2,
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None,
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'___sec12'),
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('Adding the Multiplier', 2, None, '___sec13'),
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('Setting up the Problem', 2, None, '___sec14'),
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('The problem to solve', 2, None, '___sec15'),
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('The last steps', 2, None, '___sec16'),
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('A soft classifier', 2, None, '___sec17'),
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('Soft optmization problem', 2, None, '___sec18'),
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('Kernels and non-linearity', 2, None, '___sec19'),
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('The equations', 2, None, '___sec20'),
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('The problem to solve', 2, None, '___sec21'),
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("Different kernels and Mercer's theorem", 2, None, '___sec22'),
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('The moons example', 2, None, '___sec23'),
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'___sec14'),
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('Adding the Multiplier', 2, None, '___sec15'),
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('Setting up the Problem', 2, None, '___sec16'),
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('The problem to solve', 2, None, '___sec17'),
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('The last steps', 2, None, '___sec18'),
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('A soft classifier', 2, None, '___sec19'),
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('Soft optmization problem', 2, None, '___sec20'),
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('Kernels and non-linearity', 2, None, '___sec21'),
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('The equations', 2, None, '___sec22'),
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('The problem to solve', 2, None, '___sec23'),
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("Different kernels and Mercer's theorem", 2, None, '___sec24'),
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('The moons example', 2, None, '___sec25'),
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('Mathematical optimization of convex functions',
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2,
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None,
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'___sec24'),
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('How do we solve these problems?', 2, None, '___sec25'),
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('A simple example', 2, None, '___sec26'),
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('Back to the more realistic cases', 2, None, '___sec27')]}
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'___sec26'),
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('How do we solve these problems?', 2, None, '___sec27'),
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('A simple example', 2, None, '___sec28'),
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('Back to the more realistic cases', 2, None, '___sec29')]}
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end of tocinfo -->
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<body>
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@@ -113,33 +115,35 @@ 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#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
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<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;">Getting into the details</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Thursday</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Friday</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">Hyperplanes and all that</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">What is a hyperplane?</a></li>
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<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">The two-dimensional case</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Getting into the details</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">A better approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">Setting up the Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">Soft optmization problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Kernels and non-linearity</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">The moons example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">How do we solve these problems?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs029.html#___sec28" style="font-size: 80%;">A simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs030.html#___sec29" 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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@@ -155,24 +159,44 @@ MathJax.Hub.Config({
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<a name="part0007"></a>
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<!-- !split -->
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<h2 id="___sec6" class="anchor">Getting into the details </h2>
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<h2 id="___sec6" class="anchor">A \( p \)-dimensional space of features </h2>
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<p>
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Let us define the function
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We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \).
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In a \( p \)-dimensional space of say \( p \) features we have a hyperplane defines as
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$$
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f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0,
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b+wx_1+w_2x_2+\dots +w_px_p=0.
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$$
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as the function that determines the line \( L \) that separates two classes (our two features), see the figure here.
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If we define a
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matrix \( \boldsymbol{X}=\left[\boldsymbol{x}_1,\boldsymbol{x}_2,\dots, \boldsymbol{x}_p\right] \)
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of dimension \( n\times p \), where \( n \) represents the observations for each feature and each vector \( x_i \) is a column vector of the matrix \( \boldsymbol{X} \),
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$$
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\boldsymbol{x}_i = \begin{bmatrix} x_{i1} \\ x_{i2} \\ \dots \\ \dots \\ x_{ip} \end{bmatrix}.
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$$
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If the above condition is not met for a given vector \( \boldsymbol{x}_i \) we have
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$$
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b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} >0,
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$$
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if our output \( y_i=1 \).
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In this case we say that \( \boldsymbol{x}_i \) lies on one of the sides of the hyperplane and if
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$$
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b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} < 0,
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$$
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for the class of observations \( y_i=-1 \),
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then \( \boldsymbol{x}_i \) lies on the other side.
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<p>
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Any point defined by \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_2 \) on the line \( L \) will satisfy \( \boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0 \).
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Equivalently, for the two classes of observations we have
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$$
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y_i\left(b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip}\right) > 0.
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$$
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<p>
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The signed distance \( \delta \) from any point defined by a vector \( \boldsymbol{x} \) and a point \( \boldsymbol{x}_0 \) on the line \( L \) is then
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$$
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\delta = \frac{1}{\vert\vert \boldsymbol{w}\vert\vert}(\boldsymbol{w}^T\boldsymbol{x}+b).
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$$
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When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located.
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<p>
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<p>
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@@ -197,7 +221,7 @@ $$
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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="">...</a></li>
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<li><a href="._week46-bs028.html">29</a></li>
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<li><a href="._week46-bs030.html">31</a></li>
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<li><a href="._week46-bs008.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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