mhjensen
275 lines
17 KiB
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275 lines
17 KiB
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{'highest level': 2,
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'sections': [('Support Vector Machines, overarching aims', 2, None, '___sec0'),
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('Hyperplanes and all that', 2, None, '___sec1'),
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('What is a hyperplane?', 2, None, '___sec2'),
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('A $p$-dimensional space of features', 2, None, '___sec3'),
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('The two-dimensional case', 2, None, '___sec4'),
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('Getting into the details', 2, None, '___sec5'),
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('First attempt at a minimization approach', 2, None, '___sec6'),
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('Solving the equations', 2, None, '___sec7'),
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('Problems with the Simpler Approach', 2, None, '___sec9'),
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('A quick Reminder on Lagrangian Multipliers',
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2,
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None,
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'___sec11'),
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('Adding the Multiplier', 2, None, '___sec12'),
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('Setting up the Problem', 2, None, '___sec13'),
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('The problem to solve', 2, None, '___sec14'),
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('The last steps', 2, None, '___sec15'),
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('A soft classifier', 2, None, '___sec16'),
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('Soft optmization problem', 2, None, '___sec17'),
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('Kernels and non-linearity', 2, None, '___sec18'),
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('The equations', 2, None, '___sec19'),
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('The problem to solve', 2, None, '___sec20'),
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("Different kernels and Mercer's theorem", 2, None, '___sec21'),
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('The moons example', 2, None, '___sec22'),
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('Mathematical optimization of convex functions',
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2,
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None,
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'___sec23'),
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('How do we solve these problems?', 2, None, '___sec24'),
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('A simple example', 2, None, '___sec25'),
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('Back to the more realistic cases', 2, None, '___sec26')]}
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<a class="navbar-brand" href="week46-bs.html">Week 46: Support Vector Machines</a>
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<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Hyperplanes and all that</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">What is a hyperplane?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">The two-dimensional case</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">Getting into the details</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">A better approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Setting up the Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">Soft optmization problem</a></li>
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<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Kernels and non-linearity</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The moons example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">How do we solve these problems?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">A simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Back to the more realistic cases</a></li>
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<!-- !split -->
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<h2 id="___sec18" class="anchor">Kernels and non-linearity </h2>
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<p>
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The cases we have studied till now, were all characterized by two classes
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with a close to linear separability. The classifiers we have described
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so far find linear boundaries in our input feature space. It is
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possible to make our procedure more flexible by exploring the feature
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space using other basis expansions such as higher-order polynomials,
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wavelets, splines etc.
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<p>
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If our feature space is not easy to separate, as shown in the figure
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here, we can achieve a better separation by introducing more complex
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basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to
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obtain a separation between the classes which is almost linear.
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<p>
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The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that
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we need to introduce for example a polynomial transformation to a two-dimensional training set.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
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np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">42</span>)
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<span style="color: #408080; font-style: italic"># To plot pretty figures</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'axes.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">14</span>
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'xtick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'ytick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets
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X1D <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-4</span>, <span style="color: #666666">4</span>, <span style="color: #666666">9</span>)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
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X2D <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[X1D, X1D<span style="color: #666666">**2</span>]
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">0</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>])
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plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
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plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">121</span>)
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plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
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plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
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plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">4</span>), <span style="color: #BA2121">"bs"</span>)
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plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">5</span>), <span style="color: #BA2121">"g^"</span>)
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plt<span style="color: #666666">.</span>gca()<span style="color: #666666">.</span>get_yaxis()<span style="color: #666666">.</span>set_ticks([])
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
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plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">-0.2</span>, <span style="color: #666666">0.2</span>])
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plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">122</span>)
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plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
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plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
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plt<span style="color: #666666">.</span>axvline(x<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
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plt<span style="color: #666666">.</span>plot(X2D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], X2D[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
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plt<span style="color: #666666">.</span>plot(X2D[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], X2D[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==1</span>], <span style="color: #BA2121">"g^"</span>)
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$x_2$"</span>, fontsize<span style="color: #666666">=20</span>, rotation<span style="color: #666666">=0</span>)
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plt<span style="color: #666666">.</span>gca()<span style="color: #666666">.</span>get_yaxis()<span style="color: #666666">.</span>set_ticks([<span style="color: #666666">0</span>, <span style="color: #666666">4</span>, <span style="color: #666666">8</span>, <span style="color: #666666">12</span>, <span style="color: #666666">16</span>])
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plt<span style="color: #666666">.</span>plot([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>], [<span style="color: #666666">6.5</span>, <span style="color: #666666">6.5</span>], <span style="color: #BA2121">"r--"</span>, linewidth<span style="color: #666666">=3</span>)
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plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">17</span>])
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plt<span style="color: #666666">.</span>subplots_adjust(right<span style="color: #666666">=1</span>)
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plt<span style="color: #666666">.</span>show()
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</pre></div>
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<p>
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<p>
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