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<section id="decomposing-signals-in-components-matrix-factorization-problems">
<span id="decompositions"></span><h1><span class="section-number">2.5. </span>Decomposing signals in components (matrix factorization problems)<a class="headerlink" href="#decomposing-signals-in-components-matrix-factorization-problems" title="Link to this heading">#</a></h1>
<section id="principal-component-analysis-pca">
<span id="pca"></span><h2><span class="section-number">2.5.1. </span>Principal component analysis (PCA)<a class="headerlink" href="#principal-component-analysis-pca" title="Link to this heading">#</a></h2>
<section id="exact-pca-and-probabilistic-interpretation">
<h3><span class="section-number">2.5.1.1. </span>Exact PCA and probabilistic interpretation<a class="headerlink" href="#exact-pca-and-probabilistic-interpretation" title="Link to this heading">#</a></h3>
<p>PCA is used to decompose a multivariate dataset in a set of successive
orthogonal components that explain a maximum amount of the variance. In
scikit-learn, <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> is implemented as a <em>transformer</em> object
that learns <span class="math notranslate nohighlight">\(n\)</span> components in its <code class="docutils literal notranslate"><span class="pre">fit</span></code> method, and can be used on new
data to project it on these components.</p>
<p>PCA centers but does not scale the input data for each feature before
applying the SVD. The optional parameter <code class="docutils literal notranslate"><span class="pre">whiten=True</span></code> makes it
possible to project the data onto the singular space while scaling each
component to unit variance. This is often useful if the models down-stream make
strong assumptions on the isotropy of the signal: this is for example the case
for Support Vector Machines with the RBF kernel and the K-Means clustering
algorithm.</p>
<p>Below is an example of the iris dataset, which is comprised of 4
features, projected on the 2 dimensions that explain most variance:</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_pca_vs_lda.html"><img alt="../_images/sphx_glr_plot_pca_vs_lda_001.png" src="../_images/sphx_glr_plot_pca_vs_lda_001.png" style="width: 480.0px; height: 360.0px;" />
</a>
</figure>
<p>The <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> object also provides a
probabilistic interpretation of the PCA that can give a likelihood of
data based on the amount of variance it explains. As such it implements a
<a class="reference internal" href="../glossary.html#term-score"><span class="xref std std-term">score</span></a> method that can be used in cross-validation:</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_pca_vs_fa_model_selection.html"><img alt="../_images/sphx_glr_plot_pca_vs_fa_model_selection_001.png" src="../_images/sphx_glr_plot_pca_vs_fa_model_selection_001.png" style="width: 480.0px; height: 360.0px;" />
</a>
</figure>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_pca_iris.html#sphx-glr-auto-examples-decomposition-plot-pca-iris-py"><span class="std std-ref">Principal Component Analysis (PCA) on Iris Dataset</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_pca_vs_lda.html#sphx-glr-auto-examples-decomposition-plot-pca-vs-lda-py"><span class="std std-ref">Comparison of LDA and PCA 2D projection of Iris dataset</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_pca_vs_fa_model_selection.html#sphx-glr-auto-examples-decomposition-plot-pca-vs-fa-model-selection-py"><span class="std std-ref">Model selection with Probabilistic PCA and Factor Analysis (FA)</span></a></p></li>
</ul>
</section>
<section id="incremental-pca">
<span id="incrementalpca"></span><h3><span class="section-number">2.5.1.2. </span>Incremental PCA<a class="headerlink" href="#incremental-pca" title="Link to this heading">#</a></h3>
<p>The <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> object is very useful, but has certain limitations for
large datasets. The biggest limitation is that <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> only supports
batch processing, which means all of the data to be processed must fit in main
memory. The <a class="reference internal" href="generated/sklearn.decomposition.IncrementalPCA.html#sklearn.decomposition.IncrementalPCA" title="sklearn.decomposition.IncrementalPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">IncrementalPCA</span></code></a> object uses a different form of
processing and allows for partial computations which almost
exactly match the results of <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> while processing the data in a
minibatch fashion. <a class="reference internal" href="generated/sklearn.decomposition.IncrementalPCA.html#sklearn.decomposition.IncrementalPCA" title="sklearn.decomposition.IncrementalPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">IncrementalPCA</span></code></a> makes it possible to implement
out-of-core Principal Component Analysis either by:</p>
<ul class="simple">
<li><p>Using its <code class="docutils literal notranslate"><span class="pre">partial_fit</span></code> method on chunks of data fetched sequentially
from the local hard drive or a network database.</p></li>
<li><p>Calling its fit method on a memory mapped file using
<code class="docutils literal notranslate"><span class="pre">numpy.memmap</span></code>.</p></li>
</ul>
<p><a class="reference internal" href="generated/sklearn.decomposition.IncrementalPCA.html#sklearn.decomposition.IncrementalPCA" title="sklearn.decomposition.IncrementalPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">IncrementalPCA</span></code></a> only stores estimates of component and noise variances,
in order update <code class="docutils literal notranslate"><span class="pre">explained_variance_ratio_</span></code> incrementally. This is why
memory usage depends on the number of samples per batch, rather than the
number of samples to be processed in the dataset.</p>
<p>As in <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a>, <a class="reference internal" href="generated/sklearn.decomposition.IncrementalPCA.html#sklearn.decomposition.IncrementalPCA" title="sklearn.decomposition.IncrementalPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">IncrementalPCA</span></code></a> centers but does not scale the
input data for each feature before applying the SVD.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_incremental_pca.html"><img alt="../_images/sphx_glr_plot_incremental_pca_001.png" src="../_images/sphx_glr_plot_incremental_pca_001.png" style="width: 600.0px; height: 600.0px;" />
</a>
</figure>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_incremental_pca.html"><img alt="../_images/sphx_glr_plot_incremental_pca_002.png" src="../_images/sphx_glr_plot_incremental_pca_002.png" style="width: 600.0px; height: 600.0px;" />
</a>
</figure>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_incremental_pca.html#sphx-glr-auto-examples-decomposition-plot-incremental-pca-py"><span class="std std-ref">Incremental PCA</span></a></p></li>
</ul>
</section>
<section id="pca-using-randomized-svd">
<span id="randomizedpca"></span><h3><span class="section-number">2.5.1.3. </span>PCA using randomized SVD<a class="headerlink" href="#pca-using-randomized-svd" title="Link to this heading">#</a></h3>
<p>It is often interesting to project data to a lower-dimensional
space that preserves most of the variance, by dropping the singular vector
of components associated with lower singular values.</p>
<p>For instance, if we work with 64x64 pixel gray-level pictures
for face recognition,
the dimensionality of the data is 4096 and it is slow to train an
RBF support vector machine on such wide data. Furthermore we know that
the intrinsic dimensionality of the data is much lower than 4096 since all
pictures of human faces look somewhat alike.
The samples lie on a manifold of much lower
dimension (say around 200 for instance). The PCA algorithm can be used
to linearly transform the data while both reducing the dimensionality
and preserve most of the explained variance at the same time.</p>
<p>The class <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> used with the optional parameter
<code class="docutils literal notranslate"><span class="pre">svd_solver='randomized'</span></code> is very useful in that case: since we are going
to drop most of the singular vectors it is much more efficient to limit the
computation to an approximated estimate of the singular vectors we will keep
to actually perform the transform.</p>
<p>For instance, the following shows 16 sample portraits (centered around
0.0) from the Olivetti dataset. On the right hand side are the first 16
singular vectors reshaped as portraits. Since we only require the top
16 singular vectors of a dataset with size <span class="math notranslate nohighlight">\(n_{samples} = 400\)</span>
and <span class="math notranslate nohighlight">\(n_{features} = 64 \times 64 = 4096\)</span>, the computation time is
less than 1s:</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="orig_img" src="../_images/sphx_glr_plot_faces_decomposition_001.png" style="width: 360.0px; height: 275.4px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="pca_img" src="../_images/sphx_glr_plot_faces_decomposition_002.png" style="width: 360.0px; height: 275.4px;" /></a></strong></p><p>If we note <span class="math notranslate nohighlight">\(n_{\max} = \max(n_{\mathrm{samples}}, n_{\mathrm{features}})\)</span> and
<span class="math notranslate nohighlight">\(n_{\min} = \min(n_{\mathrm{samples}}, n_{\mathrm{features}})\)</span>, the time complexity
of the randomized <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> is <span class="math notranslate nohighlight">\(O(n_{\max}^2 \cdot n_{\mathrm{components}})\)</span>
instead of <span class="math notranslate nohighlight">\(O(n_{\max}^2 \cdot n_{\min})\)</span> for the exact method
implemented in <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a>.</p>
<p>The memory footprint of randomized <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> is also proportional to
<span class="math notranslate nohighlight">\(2 \cdot n_{\max} \cdot n_{\mathrm{components}}\)</span> instead of <span class="math notranslate nohighlight">\(n_{\max}
\cdot n_{\min}\)</span> for the exact method.</p>
<p>Note: the implementation of <code class="docutils literal notranslate"><span class="pre">inverse_transform</span></code> in <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> with
<code class="docutils literal notranslate"><span class="pre">svd_solver='randomized'</span></code> is not the exact inverse transform of
<code class="docutils literal notranslate"><span class="pre">transform</span></code> even when <code class="docutils literal notranslate"><span class="pre">whiten=False</span></code> (default).</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/applications/plot_face_recognition.html#sphx-glr-auto-examples-applications-plot-face-recognition-py"><span class="std std-ref">Faces recognition example using eigenfaces and SVMs</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_faces_decomposition.html#sphx-glr-auto-examples-decomposition-plot-faces-decomposition-py"><span class="std std-ref">Faces dataset decompositions</span></a></p></li>
</ul>
<p class="rubric">References</p>
<ul class="simple">
<li><p>Algorithm 4.3 in
<a class="reference external" href="https://arxiv.org/abs/0909.4061">“Finding structure with randomness: Stochastic algorithms for
constructing approximate matrix decompositions”</a>
Halko, et al., 2009</p></li>
<li><p><a class="reference external" href="https://arxiv.org/abs/1412.3510">“An implementation of a randomized algorithm for principal component
analysis”</a> A. Szlam et al. 2014</p></li>
</ul>
</section>
<section id="sparse-principal-components-analysis-sparsepca-and-minibatchsparsepca">
<span id="sparsepca"></span><h3><span class="section-number">2.5.1.4. </span>Sparse principal components analysis (SparsePCA and MiniBatchSparsePCA)<a class="headerlink" href="#sparse-principal-components-analysis-sparsepca-and-minibatchsparsepca" title="Link to this heading">#</a></h3>
<p><a class="reference internal" href="generated/sklearn.decomposition.SparsePCA.html#sklearn.decomposition.SparsePCA" title="sklearn.decomposition.SparsePCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">SparsePCA</span></code></a> is a variant of PCA, with the goal of extracting the
set of sparse components that best reconstruct the data.</p>
<p>Mini-batch sparse PCA (<a class="reference internal" href="generated/sklearn.decomposition.MiniBatchSparsePCA.html#sklearn.decomposition.MiniBatchSparsePCA" title="sklearn.decomposition.MiniBatchSparsePCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchSparsePCA</span></code></a>) is a variant of
<a class="reference internal" href="generated/sklearn.decomposition.SparsePCA.html#sklearn.decomposition.SparsePCA" title="sklearn.decomposition.SparsePCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">SparsePCA</span></code></a> that is faster but less accurate. The increased speed is
reached by iterating over small chunks of the set of features, for a given
number of iterations.</p>
<p>Principal component analysis (<a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a>) has the disadvantage that the
components extracted by this method have exclusively dense expressions, i.e.
they have non-zero coefficients when expressed as linear combinations of the
original variables. This can make interpretation difficult. In many cases,
the real underlying components can be more naturally imagined as sparse
vectors; for example in face recognition, components might naturally map to
parts of faces.</p>
<p>Sparse principal components yields a more parsimonious, interpretable
representation, clearly emphasizing which of the original features contribute
to the differences between samples.</p>
<p>The following example illustrates 16 components extracted using sparse PCA from
the Olivetti faces dataset. It can be seen how the regularization term induces
many zeros. Furthermore, the natural structure of the data causes the non-zero
coefficients to be vertically adjacent. The model does not enforce this
mathematically: each component is a vector <span class="math notranslate nohighlight">\(h \in \mathbf{R}^{4096}\)</span>, and
there is no notion of vertical adjacency except during the human-friendly
visualization as 64x64 pixel images. The fact that the components shown below
appear local is the effect of the inherent structure of the data, which makes
such local patterns minimize reconstruction error. There exist sparsity-inducing
norms that take into account adjacency and different kinds of structure; see
<a class="reference internal" href="#jen09" id="id1"><span>[Jen09]</span></a> for a review of such methods.
For more details on how to use Sparse PCA, see the Examples section, below.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="pca_img" src="../_images/sphx_glr_plot_faces_decomposition_002.png" style="width: 360.0px; height: 275.4px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="spca_img" src="../_images/sphx_glr_plot_faces_decomposition_005.png" style="width: 360.0px; height: 275.4px;" /></a></strong></p><p>Note that there are many different formulations for the Sparse PCA
problem. The one implemented here is based on <a class="reference internal" href="#mrl09" id="id2"><span>[Mrl09]</span></a> . The optimization
problem solved is a PCA problem (dictionary learning) with an
<span class="math notranslate nohighlight">\(\ell_1\)</span> penalty on the components:</p>
<div class="math notranslate nohighlight">
\[\begin{split}(U^*, V^*) = \underset{U, V}{\operatorname{arg\,min\,}} & \frac{1}{2}
||X-UV||_{\text{Fro}}^2+\alpha||V||_{1,1} \\
\text{subject to } & ||U_k||_2 <= 1 \text{ for all }
0 \leq k < n_{components}\end{split}\]</div>
<p><span class="math notranslate nohighlight">\(||.||_{\text{Fro}}\)</span> stands for the Frobenius norm and <span class="math notranslate nohighlight">\(||.||_{1,1}\)</span>
stands for the entry-wise matrix norm which is the sum of the absolute values
of all the entries in the matrix.
The sparsity-inducing <span class="math notranslate nohighlight">\(||.||_{1,1}\)</span> matrix norm also prevents learning
components from noise when few training samples are available. The degree
of penalization (and thus sparsity) can be adjusted through the
hyperparameter <code class="docutils literal notranslate"><span class="pre">alpha</span></code>. Small values lead to a gently regularized
factorization, while larger values shrink many coefficients to zero.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>While in the spirit of an online algorithm, the class
<a class="reference internal" href="generated/sklearn.decomposition.MiniBatchSparsePCA.html#sklearn.decomposition.MiniBatchSparsePCA" title="sklearn.decomposition.MiniBatchSparsePCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchSparsePCA</span></code></a> does not implement <code class="docutils literal notranslate"><span class="pre">partial_fit</span></code> because
the algorithm is online along the features direction, not the samples
direction.</p>
</div>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_faces_decomposition.html#sphx-glr-auto-examples-decomposition-plot-faces-decomposition-py"><span class="std std-ref">Faces dataset decompositions</span></a></p></li>
</ul>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="mrl09" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">Mrl09</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://www.di.ens.fr/~fbach/mairal_icml09.pdf">“Online Dictionary Learning for Sparse Coding”</a>
J. Mairal, F. Bach, J. Ponce, G. Sapiro, 2009</p>
</div>
<div class="citation" id="jen09" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">Jen09</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://www.di.ens.fr/~fbach/sspca_AISTATS2010.pdf">“Structured Sparse Principal Component Analysis”</a>
R. Jenatton, G. Obozinski, F. Bach, 2009</p>
</div>
</div>
</section>
</section>
<section id="kernel-principal-component-analysis-kpca">
<span id="kernel-pca"></span><h2><span class="section-number">2.5.2. </span>Kernel Principal Component Analysis (kPCA)<a class="headerlink" href="#kernel-principal-component-analysis-kpca" title="Link to this heading">#</a></h2>
<section id="exact-kernel-pca">
<h3><span class="section-number">2.5.2.1. </span>Exact Kernel PCA<a class="headerlink" href="#exact-kernel-pca" title="Link to this heading">#</a></h3>
<p><a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> is an extension of PCA which achieves non-linear
dimensionality reduction through the use of kernels (see <a class="reference internal" href="metrics.html#metrics"><span class="std std-ref">Pairwise metrics, Affinities and Kernels</span></a>) <a class="reference internal" href="#scholkopf1997" id="id3"><span>[Scholkopf1997]</span></a>. It
has many applications including denoising, compression and structured
prediction (kernel dependency estimation). <a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> supports both
<code class="docutils literal notranslate"><span class="pre">transform</span></code> and <code class="docutils literal notranslate"><span class="pre">inverse_transform</span></code>.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_kernel_pca.html"><img alt="../_images/sphx_glr_plot_kernel_pca_002.png" src="../_images/sphx_glr_plot_kernel_pca_002.png" style="width: 1050.0px; height: 300.0px;" />
</a>
</figure>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p><a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA.inverse_transform" title="sklearn.decomposition.KernelPCA.inverse_transform"><code class="xref py py-meth docutils literal notranslate"><span class="pre">KernelPCA.inverse_transform</span></code></a> relies on a kernel ridge to learn the
function mapping samples from the PCA basis into the original feature
space <a class="reference internal" href="#bakir2003" id="id4"><span>[Bakir2003]</span></a>. Thus, the reconstruction obtained with
<a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA.inverse_transform" title="sklearn.decomposition.KernelPCA.inverse_transform"><code class="xref py py-meth docutils literal notranslate"><span class="pre">KernelPCA.inverse_transform</span></code></a> is an approximation. See the example
linked below for more details.</p>
</div>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_kernel_pca.html#sphx-glr-auto-examples-decomposition-plot-kernel-pca-py"><span class="std std-ref">Kernel PCA</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/applications/plot_digits_denoising.html#sphx-glr-auto-examples-applications-plot-digits-denoising-py"><span class="std std-ref">Image denoising using kernel PCA</span></a></p></li>
</ul>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="scholkopf1997" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">Scholkopf1997</a><span class="fn-bracket">]</span></span>
<p>Schölkopf, Bernhard, Alexander Smola, and Klaus-Robert Müller.
<a class="reference external" href="https://people.eecs.berkeley.edu/~wainwrig/stat241b/scholkopf_kernel.pdf">“Kernel principal component analysis.”</a>
International conference on artificial neural networks.
Springer, Berlin, Heidelberg, 1997.</p>
</div>
<div class="citation" id="bakir2003" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">Bakir2003</a><span class="fn-bracket">]</span></span>
<p>Bakır, Gökhan H., Jason Weston, and Bernhard Schölkopf.
<a class="reference external" href="https://papers.nips.cc/paper/2003/file/ac1ad983e08ad3304a97e147f522747e-Paper.pdf">“Learning to find pre-images.”</a>
Advances in neural information processing systems 16 (2003): 449-456.</p>
</div>
</div>
</section>
<section id="choice-of-solver-for-kernel-pca">
<span id="kpca-solvers"></span><h3><span class="section-number">2.5.2.2. </span>Choice of solver for Kernel PCA<a class="headerlink" href="#choice-of-solver-for-kernel-pca" title="Link to this heading">#</a></h3>
<p>While in <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> the number of components is bounded by the number of
features, in <a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> the number of components is bounded by the
number of samples. Many real-world datasets have large number of samples! In
these cases finding <em>all</em> the components with a full kPCA is a waste of
computation time, as data is mostly described by the first few components
(e.g. <code class="docutils literal notranslate"><span class="pre">n_components<=100</span></code>). In other words, the centered Gram matrix that
is eigendecomposed in the Kernel PCA fitting process has an effective rank that
is much smaller than its size. This is a situation where approximate
eigensolvers can provide speedup with very low precision loss.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="eigensolvers">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Eigensolvers<a class="headerlink" href="#eigensolvers" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The optional parameter <code class="docutils literal notranslate"><span class="pre">eigen_solver='randomized'</span></code> can be used to
<em>significantly</em> reduce the computation time when the number of requested
<code class="docutils literal notranslate"><span class="pre">n_components</span></code> is small compared with the number of samples. It relies on
randomized decomposition methods to find an approximate solution in a shorter
time.</p>
<p class="sd-card-text">The time complexity of the randomized <a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> is
<span class="math notranslate nohighlight">\(O(n_{\mathrm{samples}}^2 \cdot n_{\mathrm{components}})\)</span>
instead of <span class="math notranslate nohighlight">\(O(n_{\mathrm{samples}}^3)\)</span> for the exact method
implemented with <code class="docutils literal notranslate"><span class="pre">eigen_solver='dense'</span></code>.</p>
<p class="sd-card-text">The memory footprint of randomized <a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> is also proportional to
<span class="math notranslate nohighlight">\(2 \cdot n_{\mathrm{samples}} \cdot n_{\mathrm{components}}\)</span> instead of
<span class="math notranslate nohighlight">\(n_{\mathrm{samples}}^2\)</span> for the exact method.</p>
<p class="sd-card-text">Note: this technique is the same as in <a class="reference internal" href="#randomizedpca"><span class="std std-ref">PCA using randomized SVD</span></a>.</p>
<p class="sd-card-text">In addition to the above two solvers, <code class="docutils literal notranslate"><span class="pre">eigen_solver='arpack'</span></code> can be used as
an alternate way to get an approximate decomposition. In practice, this method
only provides reasonable execution times when the number of components to find
is extremely small. It is enabled by default when the desired number of
components is less than 10 (strict) and the number of samples is more than 200
(strict). See <a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> for details.</p>
<p class="rubric">References</p>
<ul class="simple">
<li><p class="sd-card-text"><em>dense</em> solver:
<a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.eigh.html">scipy.linalg.eigh documentation</a></p></li>
<li><p class="sd-card-text"><em>randomized</em> solver:</p>
<ul>
<li><p class="sd-card-text">Algorithm 4.3 in
<a class="reference external" href="https://arxiv.org/abs/0909.4061">“Finding structure with randomness: Stochastic
algorithms for constructing approximate matrix decompositions”</a>
Halko, et al. (2009)</p></li>
<li><p class="sd-card-text"><a class="reference external" href="https://arxiv.org/abs/1412.3510">“An implementation of a randomized algorithm
for principal component analysis”</a>
A. Szlam et al. (2014)</p></li>
</ul>
</li>
<li><p class="sd-card-text"><em>arpack</em> solver:
<a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.eigsh.html">scipy.sparse.linalg.eigsh documentation</a>
R. B. Lehoucq, D. C. Sorensen, and C. Yang, (1998)</p></li>
</ul>
</div>
</details></section>
</section>
<section id="truncated-singular-value-decomposition-and-latent-semantic-analysis">
<span id="lsa"></span><h2><span class="section-number">2.5.3. </span>Truncated singular value decomposition and latent semantic analysis<a class="headerlink" href="#truncated-singular-value-decomposition-and-latent-semantic-analysis" title="Link to this heading">#</a></h2>
<p><a class="reference internal" href="generated/sklearn.decomposition.TruncatedSVD.html#sklearn.decomposition.TruncatedSVD" title="sklearn.decomposition.TruncatedSVD"><code class="xref py py-class docutils literal notranslate"><span class="pre">TruncatedSVD</span></code></a> implements a variant of singular value decomposition
(SVD) that only computes the <span class="math notranslate nohighlight">\(k\)</span> largest singular values,
where <span class="math notranslate nohighlight">\(k\)</span> is a user-specified parameter.</p>
<p><a class="reference internal" href="generated/sklearn.decomposition.TruncatedSVD.html#sklearn.decomposition.TruncatedSVD" title="sklearn.decomposition.TruncatedSVD"><code class="xref py py-class docutils literal notranslate"><span class="pre">TruncatedSVD</span></code></a> is very similar to <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a>, but differs
in that the matrix <span class="math notranslate nohighlight">\(X\)</span> does not need to be centered.
When the columnwise (per-feature) means of <span class="math notranslate nohighlight">\(X\)</span>
are subtracted from the feature values,
truncated SVD on the resulting matrix is equivalent to PCA.</p>