mhjensen
154 lines
5.3 KiB
HTML
154 lines
5.3 KiB
HTML
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<title>Data Analysis and Machine Learning: Dimensionality Reduction</title>
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Reducing the number of degrees of freedom, overarching view',
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2,
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None,
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'___sec0'),
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('Principal Component Analysis', 2, None, '___sec1'),
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('Kernel PCA', 2, None, '___sec2'),
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('LLE', 2, None, '___sec3'),
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('Other techniques', 2, None, '___sec4')]}
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end of tocinfo -->
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<a class="navbar-brand" href="DimRed-bs.html">Data Analysis and Machine Learning: Dimensionality Reduction</a>
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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="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Principal Component Analysis</a></li>
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<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;">Kernel PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">LLE</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">Other techniques</a></li>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0003"></a>
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<!-- !split -->
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<h2 id="___sec2" 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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Kernel PCA
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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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from sklearn.decomposition import KernelPCA
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rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
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X_reduced = rbf_pca.fit_transform(X)
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Figure 8-
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<p>
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<li><a href="._DimRed-bs002.html">«</a></li>
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<li><a href="._DimRed-bs000.html">1</a></li>
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<li><a href="._DimRed-bs001.html">2</a></li>
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