further update, added also
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mhjensen
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
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None,
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'___sec14'),
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('Towards the PCA theorem', 2, None, '___sec15'),
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('Classical PCA Theorem', 2, None, '___sec16'),
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('Prof of the PCA Theorem', 2, None, '___sec17'),
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('Getting started with PCA', 2, None, '___sec18'),
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('Principal Component Analysis', 2, None, '___sec19'),
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('PCA and scikit-learn', 2, None, '___sec20'),
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('More on the PCA', 2, None, '___sec21'),
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('Incremental PCA', 2, None, '___sec22'),
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('Randomized PCA', 2, None, '___sec23'),
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('Kernel PCA', 2, None, '___sec24'),
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('LLE', 2, None, '___sec25'),
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('Other techniques', 2, None, '___sec26')]}
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('The Algorithm before the Theorem', 2, None, '___sec16'),
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('Classical PCA Theorem', 2, None, '___sec17'),
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('Prof of the PCA Theorem', 2, None, '___sec18'),
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('Getting started with PCA', 2, None, '___sec19'),
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('Principal Component Analysis', 2, None, '___sec20'),
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('PCA and scikit-learn', 2, None, '___sec21'),
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('More on the PCA', 2, None, '___sec22'),
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('Incremental PCA', 2, None, '___sec23'),
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('Randomized PCA', 2, None, '___sec24'),
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('Kernel PCA', 2, None, '___sec25'),
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('LLE', 2, None, '___sec26'),
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('Other techniques', 2, None, '___sec27')]}
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end of tocinfo -->
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<body>
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@@ -152,17 +153,18 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
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<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">LLE</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
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<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
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</ul>
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</li>
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@@ -178,14 +180,32 @@ MathJax.Hub.Config({
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<a name="part0026"></a>
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<!-- !split -->
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<h2 id="___sec25" class="anchor">LLE </h2>
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<h2 id="___sec25" class="anchor">Kernel PCA </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<p>
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Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
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(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous
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algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its
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closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where
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these local relationships are best preserved (more details shortly).
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The kernel trick is a mathematical technique that implicitly maps instances into a
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very high-dimensional space (called the feature space), enabling nonlinear classification and regression
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with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature
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space corresponds to a complex nonlinear decision boundary in the original space.
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It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear
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projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at
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preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a
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twisted manifold.
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For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an
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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">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> KernelPCA
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rbf_pca <span style="color: #666666">=</span> KernelPCA(n_components <span style="color: #666666">=</span> <span style="color: #666666">2</span>, kernel<span style="color: #666666">=</span><span style="color: #BA2121">"rbf"</span>, gamma<span style="color: #666666">=0.04</span>)
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X_reduced <span style="color: #666666">=</span> rbf_pca<span style="color: #666666">.</span>fit_transform(X)
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</pre></div>
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<p>
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</div>
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</div>
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<p>
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<p>
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@@ -204,6 +224,7 @@ these local relationships are best preserved (more details shortly).
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<li><a href="._DimRed-bs025.html">26</a></li>
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<li class="active"><a href="._DimRed-bs026.html">27</a></li>
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<li><a href="._DimRed-bs027.html">28</a></li>
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<li><a href="._DimRed-bs028.html">29</a></li>
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<li><a href="._DimRed-bs027.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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