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<li><a class="reference internal" href="#">2.5. Decomposing signals in components (matrix factorization problems)</a><ul>
<li><a class="reference internal" href="#principal-component-analysis-pca">2.5.1. Principal component analysis (PCA)</a><ul>
<li><a class="reference internal" href="#exact-pca-and-probabilistic-interpretation">2.5.1.1. Exact PCA and probabilistic interpretation</a></li>
<li><a class="reference internal" href="#incremental-pca">2.5.1.2. Incremental PCA</a></li>
<li><a class="reference internal" href="#pca-using-randomized-svd">2.5.1.3. PCA using randomized SVD</a></li>
<li><a class="reference internal" href="#sparse-principal-components-analysis-sparsepca-and-minibatchsparsepca">2.5.1.4. Sparse principal components analysis (SparsePCA and MiniBatchSparsePCA)</a></li>
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<li><a class="reference internal" href="#exact-kernel-pca">2.5.2.1. Exact Kernel PCA</a></li>
<li><a class="reference internal" href="#choice-of-solver-for-kernel-pca">2.5.2.2. Choice of solver for Kernel PCA</a></li>
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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="Permalink 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="Permalink 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="Permalink 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>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<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>
</aside>
</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="Permalink 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>
<blockquote>
<div><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 sparse matrix or a memory mapped file using
<code class="docutils literal notranslate"><span class="pre">numpy.memmap</span></code>.</p></li>
</ul>
</div></blockquote>
<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>
<aside class="topic">
<p class="topic-title">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>
</aside>
</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="Permalink 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>
<aside class="topic">
<p class="topic-title">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>
</aside>
<aside class="topic">
<p class="topic-title">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>
</aside>
</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="Permalink 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>
<aside class="topic">
<p class="topic-title">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>
</aside>
<aside class="topic">
<p class="topic-title">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/sierra/pdfs/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>
</aside>
</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="Permalink 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="Permalink 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>
<aside class="topic">
<p class="topic-title">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>
</ul>
</aside>
<aside class="topic">
<p class="topic-title">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>
</aside>
</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="Permalink 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>
<p>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>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>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>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>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>
<aside class="topic">
<p class="topic-title">References:</p>
<ul>
<li><p><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><em>randomized</em> solver:</p>
<blockquote>
<div><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>
</div></blockquote>
</li>
<li><p><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>
</aside>
</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="Permalink 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>When truncated SVD is applied to term-document matrices
(as returned by <a class="reference internal" href="generated/sklearn.feature_extraction.text.CountVectorizer.html#sklearn.feature_extraction.text.CountVectorizer" title="sklearn.feature_extraction.text.CountVectorizer"><code class="xref py py-class docutils literal notranslate"><span class="pre">CountVectorizer</span></code></a> or
<a class="reference internal" href="generated/sklearn.feature_extraction.text.TfidfVectorizer.html#sklearn.feature_extraction.text.TfidfVectorizer" title="sklearn.feature_extraction.text.TfidfVectorizer"><code class="xref py py-class docutils literal notranslate"><span class="pre">TfidfVectorizer</span></code></a>),
this transformation is known as
<a class="reference external" href="https://nlp.stanford.edu/IR-book/pdf/18lsi.pdf">latent semantic analysis</a>
(LSA), because it transforms such matrices
to a “semantic” space of low dimensionality.
In particular, LSA is known to combat the effects of synonymy and polysemy
(both of which roughly mean there are multiple meanings per word),
which cause term-document matrices to be overly sparse
and exhibit poor similarity under measures such as cosine similarity.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>LSA is also known as latent semantic indexing, LSI,
though strictly that refers to its use in persistent indexes
for information retrieval purposes.</p>
</div>
<p>Mathematically, truncated SVD applied to training samples <span class="math notranslate nohighlight">\(X\)</span>
produces a low-rank approximation <span class="math notranslate nohighlight">\(X\)</span>:</p>
<div class="math notranslate nohighlight">
\[X \approx X_k = U_k \Sigma_k V_k^\top\]</div>
<p>After this operation, <span class="math notranslate nohighlight">\(U_k \Sigma_k\)</span>
is the transformed training set with <span class="math notranslate nohighlight">\(k\)</span> features
(called <code class="docutils literal notranslate"><span class="pre">n_components</span></code> in the API).</p>
<p>To also transform a test set <span class="math notranslate nohighlight">\(X\)</span>, we multiply it with <span class="math notranslate nohighlight">\(V_k\)</span>:</p>
<div class="math notranslate nohighlight">
\[X' = X V_k\]</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Most treatments of LSA in the natural language processing (NLP)
and information retrieval (IR) literature
swap the axes of the matrix <span class="math notranslate nohighlight">\(X\)</span> so that it has shape
<code class="docutils literal notranslate"><span class="pre">n_features</span></code> × <code class="docutils literal notranslate"><span class="pre">n_samples</span></code>.
We present LSA in a different way that matches the scikit-learn API better,
but the singular values found are the same.</p>
</div>
<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.
In practical terms, this means
that the <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> transformer accepts <code class="docutils literal notranslate"><span class="pre">scipy.sparse</span></code>
matrices without the need to densify them,
as densifying may fill up memory even for medium-sized document collections.</p>
<p>While the <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> transformer
works with any feature matrix,
using it on tf–idf matrices is recommended over raw frequency counts
in an LSA/document processing setting.
In particular, sublinear scaling and inverse document frequency
should be turned on (<code class="docutils literal notranslate"><span class="pre">sublinear_tf=True,</span> <span class="pre">use_idf=True</span></code>)
to bring the feature values closer to a Gaussian distribution,
compensating for LSA’s erroneous assumptions about textual data.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/text/plot_document_clustering.html#sphx-glr-auto-examples-text-plot-document-clustering-py"><span class="std std-ref">Clustering text documents using k-means</span></a></p></li>
</ul>
</aside>
<aside class="topic">
<p class="topic-title">References:</p>
<ul class="simple">
<li><p>Christopher D. Manning, Prabhakar Raghavan and Hinrich Schütze (2008),
<em>Introduction to Information Retrieval</em>, Cambridge University Press,
chapter 18: <a class="reference external" href="https://nlp.stanford.edu/IR-book/pdf/18lsi.pdf">Matrix decompositions & latent semantic indexing</a></p></li>
</ul>
</aside>
</section>
<section id="dictionary-learning">
<span id="dictionarylearning"></span><h2><span class="section-number">2.5.4. </span>Dictionary Learning<a class="headerlink" href="#dictionary-learning" title="Permalink to this heading">¶</a></h2>
<section id="sparse-coding-with-a-precomputed-dictionary">
<span id="sparsecoder"></span><h3><span class="section-number">2.5.4.1. </span>Sparse coding with a precomputed dictionary<a class="headerlink" href="#sparse-coding-with-a-precomputed-dictionary" title="Permalink to this heading">¶</a></h3>
<p>The <a class="reference internal" href="generated/sklearn.decomposition.SparseCoder.html#sklearn.decomposition.SparseCoder" title="sklearn.decomposition.SparseCoder"><code class="xref py py-class docutils literal notranslate"><span class="pre">SparseCoder</span></code></a> object is an estimator that can be used to transform signals
into sparse linear combination of atoms from a fixed, precomputed dictionary
such as a discrete wavelet basis. This object therefore does not
implement a <code class="docutils literal notranslate"><span class="pre">fit</span></code> method. The transformation amounts
to a sparse coding problem: finding a representation of the data as a linear
combination of as few dictionary atoms as possible. All variations of
dictionary learning implement the following transform methods, controllable via
the <code class="docutils literal notranslate"><span class="pre">transform_method</span></code> initialization parameter:</p>
<ul class="simple">
<li><p>Orthogonal matching pursuit (<a class="reference internal" href="linear_model.html#omp"><span class="std std-ref">Orthogonal Matching Pursuit (OMP)</span></a>)</p></li>
<li><p>Least-angle regression (<a class="reference internal" href="linear_model.html#least-angle-regression"><span class="std std-ref">Least Angle Regression</span></a>)</p></li>
<li><p>Lasso computed by least-angle regression</p></li>
<li><p>Lasso using coordinate descent (<a class="reference internal" href="linear_model.html#lasso"><span class="std std-ref">Lasso</span></a>)</p></li>
<li><p>Thresholding</p></li>
</ul>
<p>Thresholding is very fast but it does not yield accurate reconstructions.
They have been shown useful in literature for classification tasks. For image
reconstruction tasks, orthogonal matching pursuit yields the most accurate,
unbiased reconstruction.</p>
<p>The dictionary learning objects offer, via the <code class="docutils literal notranslate"><span class="pre">split_code</span></code> parameter, the
possibility to separate the positive and negative values in the results of
sparse coding. This is useful when dictionary learning is used for extracting
features that will be used for supervised learning, because it allows the
learning algorithm to assign different weights to negative loadings of a
particular atom, from to the corresponding positive loading.</p>
<p>The split code for a single sample has length <code class="docutils literal notranslate"><span class="pre">2</span> <span class="pre">*</span> <span class="pre">n_components</span></code>
and is constructed using the following rule: First, the regular code of length
<code class="docutils literal notranslate"><span class="pre">n_components</span></code> is computed. Then, the first <code class="docutils literal notranslate"><span class="pre">n_components</span></code> entries of the
<code class="docutils literal notranslate"><span class="pre">split_code</span></code> are
filled with the positive part of the regular code vector. The second half of
the split code is filled with the negative part of the code vector, only with
a positive sign. Therefore, the split_code is non-negative.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_sparse_coding.html#sphx-glr-auto-examples-decomposition-plot-sparse-coding-py"><span class="std std-ref">Sparse coding with a precomputed dictionary</span></a></p></li>
</ul>
</aside>
</section>
<section id="generic-dictionary-learning">
<h3><span class="section-number">2.5.4.2. </span>Generic dictionary learning<a class="headerlink" href="#generic-dictionary-learning" title="Permalink to this heading">¶</a></h3>
<p>Dictionary learning (<a class="reference internal" href="generated/sklearn.decomposition.DictionaryLearning.html#sklearn.decomposition.DictionaryLearning" title="sklearn.decomposition.DictionaryLearning"><code class="xref py py-class docutils literal notranslate"><span class="pre">DictionaryLearning</span></code></a>) is a matrix factorization
problem that amounts to finding a (usually overcomplete) dictionary that will
perform well at sparsely encoding the fitted data.</p>
<p>Representing data as sparse combinations of atoms from an overcomplete
dictionary is suggested to be the way the mammalian primary visual cortex works.
Consequently, dictionary learning applied on image patches has been shown to
give good results in image processing tasks such as image completion,
inpainting and denoising, as well as for supervised recognition tasks.</p>
<p>Dictionary learning is an optimization problem solved by alternatively updating
the sparse code, as a solution to multiple Lasso problems, considering the
dictionary fixed, and then updating the dictionary to best fit the sparse code.</p>
<div class="math notranslate nohighlight">
\[\begin{split}(U^*, V^*) = \underset{U, V}{\operatorname{arg\,min\,}} & \frac{1}{2}
||X-UV||_{\text{Fro}}^2+\alpha||U||_{1,1} \\
\text{subject to } & ||V_k||_2 <= 1 \text{ for all }
0 \leq k < n_{\mathrm{atoms}}\end{split}\]</div>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="pca_img2" 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="dict_img2" src="../_images/sphx_glr_plot_faces_decomposition_007.png" style="width: 360.0px; height: 275.4px;" /></a></strong></p><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.
After using such a procedure to fit the dictionary, the transform is simply a
sparse coding step that shares the same implementation with all dictionary
learning objects (see <a class="reference internal" href="#sparsecoder"><span class="std std-ref">Sparse coding with a precomputed dictionary</span></a>).</p>
<p>It is also possible to constrain the dictionary and/or code to be positive to
match constraints that may be present in the data. Below are the faces with
different positivity constraints applied. Red indicates negative values, blue
indicates positive values, and white represents zeros.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/decomposition/plot_image_denoising.html"><img alt="dict_img_pos1" src="../_images/sphx_glr_plot_faces_decomposition_010.png" style="width: 360.0px; height: 275.4px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_image_denoising.html"><img alt="dict_img_pos2" src="../_images/sphx_glr_plot_faces_decomposition_011.png" style="width: 360.0px; height: 275.4px;" /></a></strong></p><p class="centered">
<strong><a class="reference external" href="../auto_examples/decomposition/plot_image_denoising.html"><img alt="dict_img_pos3" src="../_images/sphx_glr_plot_faces_decomposition_012.png" style="width: 360.0px; height: 275.4px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_image_denoising.html"><img alt="dict_img_pos4" src="../_images/sphx_glr_plot_faces_decomposition_013.png" style="width: 360.0px; height: 275.4px;" /></a></strong></p><p>The following image shows how a dictionary learned from 4x4 pixel image patches
extracted from part of the image of a raccoon face looks like.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_image_denoising.html"><img alt="../_images/sphx_glr_plot_image_denoising_001.png" src="../_images/sphx_glr_plot_image_denoising_001.png" style="width: 250.0px; height: 165.0px;" /></a>
</figure>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_image_denoising.html#sphx-glr-auto-examples-decomposition-plot-image-denoising-py"><span class="std std-ref">Image denoising using dictionary learning</span></a></p></li>
</ul>
</aside>
<aside class="topic">
<p class="topic-title">References:</p>
<ul class="simple">
<li><p><a class="reference external" href="https://www.di.ens.fr/sierra/pdfs/icml09.pdf">“Online dictionary learning for sparse coding”</a>
J. Mairal, F. Bach, J. Ponce, G. Sapiro, 2009</p></li>
</ul>
</aside>
</section>
<section id="mini-batch-dictionary-learning">
<span id="minibatchdictionarylearning"></span><h3><span class="section-number">2.5.4.3. </span>Mini-batch dictionary learning<a class="headerlink" href="#mini-batch-dictionary-learning" title="Permalink to this heading">¶</a></h3>
<p><a class="reference internal" href="generated/sklearn.decomposition.MiniBatchDictionaryLearning.html#sklearn.decomposition.MiniBatchDictionaryLearning" title="sklearn.decomposition.MiniBatchDictionaryLearning"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchDictionaryLearning</span></code></a> implements a faster, but less accurate
version of the dictionary learning algorithm that is better suited for large
datasets.</p>
<p>By default, <a class="reference internal" href="generated/sklearn.decomposition.MiniBatchDictionaryLearning.html#sklearn.decomposition.MiniBatchDictionaryLearning" title="sklearn.decomposition.MiniBatchDictionaryLearning"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchDictionaryLearning</span></code></a> divides the data into
mini-batches and optimizes in an online manner by cycling over the mini-batches
for the specified number of iterations. However, at the moment it does not
implement a stopping condition.</p>
<p>The estimator also implements <code class="docutils literal notranslate"><span class="pre">partial_fit</span></code>, which updates the dictionary by
iterating only once over a mini-batch. This can be used for online learning
when the data is not readily available from the start, or for when the data
does not fit into the memory.</p>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_dict_face_patches.html"><img alt="../_images/sphx_glr_plot_dict_face_patches_001.png" class="align-right" src="../_images/sphx_glr_plot_dict_face_patches_001.png" style="width: 210.0px; height: 200.0px;" /></a>
<aside class="topic">
<p class="topic-title"><strong>Clustering for dictionary learning</strong></p>
<p>Note that when using dictionary learning to extract a representation
(e.g. for sparse coding) clustering can be a good proxy to learn the
dictionary. For instance the <a class="reference internal" href="generated/sklearn.cluster.MiniBatchKMeans.html#sklearn.cluster.MiniBatchKMeans" title="sklearn.cluster.MiniBatchKMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchKMeans</span></code></a> estimator is
computationally efficient and implements on-line learning with a
<code class="docutils literal notranslate"><span class="pre">partial_fit</span></code> method.</p>
<blockquote>
<div><p>Example: <a class="reference internal" href="../auto_examples/cluster/plot_dict_face_patches.html#sphx-glr-auto-examples-cluster-plot-dict-face-patches-py"><span class="std std-ref">Online learning of a dictionary of parts of faces</span></a></p>
</div></blockquote>
</aside>
</section>
</section>
<section id="factor-analysis">
<span id="fa"></span><h2><span class="section-number">2.5.5. </span>Factor Analysis<a class="headerlink" href="#factor-analysis" title="Permalink to this heading">¶</a></h2>
<p>In unsupervised learning we only have a dataset <span class="math notranslate nohighlight">\(X = \{x_1, x_2, \dots, x_n
\}\)</span>. How can this dataset be described mathematically? A very simple
<code class="docutils literal notranslate"><span class="pre">continuous</span> <span class="pre">latent</span> <span class="pre">variable</span></code> model for <span class="math notranslate nohighlight">\(X\)</span> is</p>
<div class="math notranslate nohighlight">
\[x_i = W h_i + \mu + \epsilon\]</div>
<p>The vector <span class="math notranslate nohighlight">\(h_i\)</span> is called “latent” because it is unobserved. <span class="math notranslate nohighlight">\(\epsilon\)</span> is
considered a noise term distributed according to a Gaussian with mean 0 and
covariance <span class="math notranslate nohighlight">\(\Psi\)</span> (i.e. <span class="math notranslate nohighlight">\(\epsilon \sim \mathcal{N}(0, \Psi)\)</span>), <span class="math notranslate nohighlight">\(\mu\)</span> is some
arbitrary offset vector. Such a model is called “generative” as it describes
how <span class="math notranslate nohighlight">\(x_i\)</span> is generated from <span class="math notranslate nohighlight">\(h_i\)</span>. If we use all the <span class="math notranslate nohighlight">\(x_i\)</span>’s as columns to form
a matrix <span class="math notranslate nohighlight">\(\mathbf{X}\)</span> and all the <span class="math notranslate nohighlight">\(h_i\)</span>’s as columns of a matrix <span class="math notranslate nohighlight">\(\mathbf{H}\)</span>
then we can write (with suitably defined <span class="math notranslate nohighlight">\(\mathbf{M}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{E}\)</span>):</p>
<div class="math notranslate nohighlight">
\[\mathbf{X} = W \mathbf{H} + \mathbf{M} + \mathbf{E}\]</div>
<p>In other words, we <em>decomposed</em> matrix <span class="math notranslate nohighlight">\(\mathbf{X}\)</span>.</p>
<p>If <span class="math notranslate nohighlight">\(h_i\)</span> is given, the above equation automatically implies the following
probabilistic interpretation:</p>
<div class="math notranslate nohighlight">
\[p(x_i|h_i) = \mathcal{N}(Wh_i + \mu, \Psi)\]</div>
<p>For a complete probabilistic model we also need a prior distribution for the
latent variable <span class="math notranslate nohighlight">\(h\)</span>. The most straightforward assumption (based on the nice
properties of the Gaussian distribution) is <span class="math notranslate nohighlight">\(h \sim \mathcal{N}(0,
\mathbf{I})\)</span>. This yields a Gaussian as the marginal distribution of <span class="math notranslate nohighlight">\(x\)</span>:</p>
<div class="math notranslate nohighlight">
\[p(x) = \mathcal{N}(\mu, WW^T + \Psi)\]</div>
<p>Now, without any further assumptions the idea of having a latent variable <span class="math notranslate nohighlight">\(h\)</span>
would be superfluous – <span class="math notranslate nohighlight">\(x\)</span> can be completely modelled with a mean
and a covariance. We need to impose some more specific structure on one
of these two parameters. A simple additional assumption regards the
structure of the error covariance <span class="math notranslate nohighlight">\(\Psi\)</span>:</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\Psi = \sigma^2 \mathbf{I}\)</span>: This assumption leads to
the probabilistic model 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>.</p></li>
<li><p><span class="math notranslate nohighlight">\(\Psi = \mathrm{diag}(\psi_1, \psi_2, \dots, \psi_n)\)</span>: This model is called
<a class="reference internal" href="generated/sklearn.decomposition.FactorAnalysis.html#sklearn.decomposition.FactorAnalysis" title="sklearn.decomposition.FactorAnalysis"><code class="xref py py-class docutils literal notranslate"><span class="pre">FactorAnalysis</span></code></a>, a classical statistical model. The matrix W is
sometimes called the “factor loading matrix”.</p></li>
</ul>
<p>Both models essentially estimate a Gaussian with a low-rank covariance matrix.
Because both models are probabilistic they can be integrated in more complex
models, e.g. Mixture of Factor Analysers. One gets very different models (e.g.
<a class="reference internal" href="generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA" title="sklearn.decomposition.FastICA"><code class="xref py py-class docutils literal notranslate"><span class="pre">FastICA</span></code></a>) if non-Gaussian priors on the latent variables are assumed.</p>
<p>Factor analysis <em>can</em> produce similar components (the columns of its loading
matrix) 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>. However, one can not make any general statements
about these components (e.g. whether they are orthogonal):</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="pca_img3" 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="fa_img3" src="../_images/sphx_glr_plot_faces_decomposition_008.png" style="width: 360.0px; height: 275.4px;" /></a></strong></p><p>The main advantage for Factor Analysis over <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 that
it can model the variance in every direction of the input space independently
(heteroscedastic noise):</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="../_images/sphx_glr_plot_faces_decomposition_009.png" src="../_images/sphx_glr_plot_faces_decomposition_009.png" style="width: 240.0px; height: 270.0px;" /></a>
</figure>
<p>This allows better model selection than probabilistic PCA in the presence
of heteroscedastic noise:</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_002.png" src="../_images/sphx_glr_plot_pca_vs_fa_model_selection_002.png" style="width: 480.0px; height: 360.0px;" /></a>
</figure>
<p>Factor Analysis is often followed by a rotation of the factors (with the
parameter <code class="docutils literal notranslate"><span class="pre">rotation</span></code>), usually to improve interpretability. For example,
Varimax rotation maximizes the sum of the variances of the squared loadings,
i.e., it tends to produce sparser factors, which are influenced by only a few
features each (the “simple structure”). See e.g., the first example below.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_varimax_fa.html#sphx-glr-auto-examples-decomposition-plot-varimax-fa-py"><span class="std std-ref">Factor Analysis (with rotation) to visualize patterns</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>
</aside>
</section>
<section id="independent-component-analysis-ica">
<span id="ica"></span><h2><span class="section-number">2.5.6. </span>Independent component analysis (ICA)<a class="headerlink" href="#independent-component-analysis-ica" title="Permalink to this heading">¶</a></h2>
<p>Independent component analysis separates a multivariate signal into
additive subcomponents that are maximally independent. It is
implemented in scikit-learn using the <a class="reference internal" href="generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA" title="sklearn.decomposition.FastICA"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fast</span> <span class="pre">ICA</span></code></a>
algorithm. Typically, ICA is not used for reducing dimensionality but
for separating superimposed signals. Since the ICA model does not include
a noise term, for the model to be correct, whitening must be applied.
This can be done internally using the whiten argument or manually using one
of the PCA variants.</p>
<p>It is classically used to separate mixed signals (a problem known as
<em>blind source separation</em>), as in the example below:</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_ica_blind_source_separation.html"><img alt="../_images/sphx_glr_plot_ica_blind_source_separation_001.png" src="../_images/sphx_glr_plot_ica_blind_source_separation_001.png" style="width: 384.0px; height: 288.0px;" /></a>
</figure>
<p>ICA can also be used as yet another non linear decomposition that finds
components with some sparsity:</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="pca_img4" 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="ica_img4" src="../_images/sphx_glr_plot_faces_decomposition_004.png" style="width: 360.0px; height: 275.4px;" /></a></strong></p><aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_ica_blind_source_separation.html#sphx-glr-auto-examples-decomposition-plot-ica-blind-source-separation-py"><span class="std std-ref">Blind source separation using FastICA</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_ica_vs_pca.html#sphx-glr-auto-examples-decomposition-plot-ica-vs-pca-py"><span class="std std-ref">FastICA on 2D point clouds</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>
</aside>
</section>
<section id="non-negative-matrix-factorization-nmf-or-nnmf">
<span id="nmf"></span><h2><span class="section-number">2.5.7. </span>Non-negative matrix factorization (NMF or NNMF)<a class="headerlink" href="#non-negative-matrix-factorization-nmf-or-nnmf" title="Permalink to this heading">¶</a></h2>
<section id="nmf-with-the-frobenius-norm">
<h3><span class="section-number">2.5.7.1. </span>NMF with the Frobenius norm<a class="headerlink" href="#nmf-with-the-frobenius-norm" title="Permalink to this heading">¶</a></h3>
<p><a class="reference internal" href="generated/sklearn.decomposition.NMF.html#sklearn.decomposition.NMF" title="sklearn.decomposition.NMF"><code class="xref py py-class docutils literal notranslate"><span class="pre">NMF</span></code></a> <a class="footnote-reference brackets" href="#id13" id="id6" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a> is an alternative approach to decomposition that assumes that the
data and the components are non-negative. <a class="reference internal" href="generated/sklearn.decomposition.NMF.html#sklearn.decomposition.NMF" title="sklearn.decomposition.NMF"><code class="xref py py-class docutils literal notranslate"><span class="pre">NMF</span></code></a> can be plugged in
instead 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> or its variants, in the cases where the data matrix
does not contain negative values. It finds a decomposition of samples
<span class="math notranslate nohighlight">\(X\)</span> into two matrices <span class="math notranslate nohighlight">\(W\)</span> and <span class="math notranslate nohighlight">\(H\)</span> of non-negative elements,
by optimizing the distance <span class="math notranslate nohighlight">\(d\)</span> between <span class="math notranslate nohighlight">\(X\)</span> and the matrix product
<span class="math notranslate nohighlight">\(WH\)</span>. The most widely used distance function is the squared Frobenius
norm, which is an obvious extension of the Euclidean norm to matrices:</p>
<div class="math notranslate nohighlight">
\[d_{\mathrm{Fro}}(X, Y) = \frac{1}{2} ||X - Y||_{\mathrm{Fro}}^2 = \frac{1}{2} \sum_{i,j} (X_{ij} - {Y}_{ij})^2\]</div>
<p>Unlike <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 representation of a vector is obtained in an additive
fashion, by superimposing the components, without subtracting. Such additive
models are efficient for representing images and text.</p>
<p>It has been observed in [Hoyer, 2004] <a class="footnote-reference brackets" href="#id14" id="id7" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a> that, when carefully constrained,
<a class="reference internal" href="generated/sklearn.decomposition.NMF.html#sklearn.decomposition.NMF" title="sklearn.decomposition.NMF"><code class="xref py py-class docutils literal notranslate"><span class="pre">NMF</span></code></a> can produce a parts-based representation of the dataset,
resulting in interpretable models. The following example displays 16
sparse components found by <a class="reference internal" href="generated/sklearn.decomposition.NMF.html#sklearn.decomposition.NMF" title="sklearn.decomposition.NMF"><code class="xref py py-class docutils literal notranslate"><span class="pre">NMF</span></code></a> from the images in the Olivetti
faces dataset, in comparison with the PCA eigenfaces.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="pca_img5" 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="nmf_img5" src="../_images/sphx_glr_plot_faces_decomposition_003.png" style="width: 360.0px; height: 275.4px;" /></a></strong></p><p>The <code class="docutils literal notranslate"><span class="pre">init</span></code> attribute determines the initialization method applied, which
has a great impact on the performance of the method. <a class="reference internal" href="generated/sklearn.decomposition.NMF.html#sklearn.decomposition.NMF" title="sklearn.decomposition.NMF"><code class="xref py py-class docutils literal notranslate"><span class="pre">NMF</span></code></a> implements the
method Nonnegative Double Singular Value Decomposition. NNDSVD <a class="footnote-reference brackets" href="#id15" id="id8" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a> is based on
two SVD processes, one approximating the data matrix, the other approximating
positive sections of the resulting partial SVD factors utilizing an algebraic
property of unit rank matrices. The basic NNDSVD algorithm is better fit for
sparse factorization. Its variants NNDSVDa (in which all zeros are set equal to
the mean of all elements of the data), and NNDSVDar (in which the zeros are set
to random perturbations less than the mean of the data divided by 100) are
recommended in the dense case.</p>
<p>Note that the Multiplicative Update (‘mu’) solver cannot update zeros present in
the initialization, so it leads to poorer results when used jointly with the
basic NNDSVD algorithm which introduces a lot of zeros; in this case, NNDSVDa or
NNDSVDar should be preferred.</p>
<p><a class="reference internal" href="generated/sklearn.decomposition.NMF.html#sklearn.decomposition.NMF" title="sklearn.decomposition.NMF"><code class="xref py py-class docutils literal notranslate"><span class="pre">NMF</span></code></a> can also be initialized with correctly scaled random non-negative
matrices by setting <code class="docutils literal notranslate"><span class="pre">init="random"</span></code>. An integer seed or a
<code class="docutils literal notranslate"><span class="pre">RandomState</span></code> can also be passed to <code class="docutils literal notranslate"><span class="pre">random_state</span></code> to control
reproducibility.</p>
<p>In <a class="reference internal" href="generated/sklearn.decomposition.NMF.html#sklearn.decomposition.NMF" title="sklearn.decomposition.NMF"><code class="xref py py-class docutils literal notranslate"><span class="pre">NMF</span></code></a>, L1 and L2 priors can be added to the loss function in order to
regularize the model. The L2 prior uses the Frobenius norm, while the L1 prior
uses an elementwise L1 norm. As in <a class="reference internal" href="generated/sklearn.linear_model.ElasticNet.html#sklearn.linear_model.ElasticNet" title="sklearn.linear_model.ElasticNet"><code class="xref py py-class docutils literal notranslate"><span class="pre">ElasticNet</span></code></a>,
we control the combination of L1 and L2 with the <code class="docutils literal notranslate"><span class="pre">l1_ratio</span></code> (<span class="math notranslate nohighlight">\(\rho\)</span>)
parameter, and the intensity of the regularization with the <code class="docutils literal notranslate"><span class="pre">alpha_W</span></code> and
<code class="docutils literal notranslate"><span class="pre">alpha_H</span></code> (<span class="math notranslate nohighlight">\(\alpha_W\)</span> and <span class="math notranslate nohighlight">\(\alpha_H\)</span>) parameters. The priors are
scaled by the number of samples (<span class="math notranslate nohighlight">\(n\_samples\)</span>) for <code class="docutils literal notranslate"><span class="pre">H</span></code> and the number of
features (<span class="math notranslate nohighlight">\(n\_features\)</span>) for <code class="docutils literal notranslate"><span class="pre">W</span></code> to keep their impact balanced with
respect to one another and to the data fit term as independent as possible of
the size of the training set. Then the priors terms are:</p>
<div class="math notranslate nohighlight">
\[(\alpha_W \rho ||W||_1 + \frac{\alpha_W(1-\rho)}{2} ||W||_{\mathrm{Fro}} ^ 2) * n\_features
+ (\alpha_H \rho ||H||_1 + \frac{\alpha_H(1-\rho)}{2} ||H||_{\mathrm{Fro}} ^ 2) * n\_samples\]</div>
<p>and the regularized objective function is:</p>
<div class="math notranslate nohighlight">
\[d_{\mathrm{Fro}}(X, WH)
+ (\alpha_W \rho ||W||_1 + \frac{\alpha_W(1-\rho)}{2} ||W||_{\mathrm{Fro}} ^ 2) * n\_features
+ (\alpha_H \rho ||H||_1 + \frac{\alpha_H(1-\rho)}{2} ||H||_{\mathrm{Fro}} ^ 2) * n\_samples\]</div>
</section>
<section id="nmf-with-a-beta-divergence">
<h3><span class="section-number">2.5.7.2. </span>NMF with a beta-divergence<a class="headerlink" href="#nmf-with-a-beta-divergence" title="Permalink to this heading">¶</a></h3>
<p>As described previously, the most widely used distance function is the squared
Frobenius norm, which is an obvious extension of the Euclidean norm to
matrices:</p>
<div class="math notranslate nohighlight">
\[d_{\mathrm{Fro}}(X, Y) = \frac{1}{2} ||X - Y||_{Fro}^2 = \frac{1}{2} \sum_{i,j} (X_{ij} - {Y}_{ij})^2\]</div>
<p>Other distance functions can be used in NMF as, for example, the (generalized)
Kullback-Leibler (KL) divergence, also referred as I-divergence:</p>
<div class="math notranslate nohighlight">
\[d_{KL}(X, Y) = \sum_{i,j} (X_{ij} \log(\frac{X_{ij}}{Y_{ij}}) - X_{ij} + Y_{ij})\]</div>
<p>Or, the Itakura-Saito (IS) divergence:</p>
<div class="math notranslate nohighlight">
\[d_{IS}(X, Y) = \sum_{i,j} (\frac{X_{ij}}{Y_{ij}} - \log(\frac{X_{ij}}{Y_{ij}}) - 1)\]</div>
<p>These three distances are special cases of the beta-divergence family, with
<span class="math notranslate nohighlight">\(\beta = 2, 1, 0\)</span> respectively <a class="footnote-reference brackets" href="#id17" id="id9" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a>. The beta-divergence are
defined by :</p>
<div class="math notranslate nohighlight">
\[d_{\beta}(X, Y) = \sum_{i,j} \frac{1}{\beta(\beta - 1)}(X_{ij}^\beta + (\beta-1)Y_{ij}^\beta - \beta X_{ij} Y_{ij}^{\beta - 1})\]</div>
<a class="reference internal image-reference" href="../_images/beta_divergence.png"><img alt="../_images/beta_divergence.png" class="align-center" src="../_images/beta_divergence.png" style="width: 480.0px; height: 360.0px;" /></a>
<p>Note that this definition is not valid if <span class="math notranslate nohighlight">\(\beta \in (0; 1)\)</span>, yet it can
be continuously extended to the definitions of <span class="math notranslate nohighlight">\(d_{KL}\)</span> and <span class="math notranslate nohighlight">\(d_{IS}\)</span>
respectively.</p>
<p><a class="reference internal" href="generated/sklearn.decomposition.NMF.html#sklearn.decomposition.NMF" title="sklearn.decomposition.NMF"><code class="xref py py-class docutils literal notranslate"><span class="pre">NMF</span></code></a> implements two solvers, using Coordinate Descent (‘cd’) <a class="footnote-reference brackets" href="#id16" id="id10" role="doc-noteref"><span class="fn-bracket">[</span>5<span class="fn-bracket">]</span></a>, and
Multiplicative Update (‘mu’) <a class="footnote-reference brackets" href="#id17" id="id11" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a>. The ‘mu’ solver can optimize every
beta-divergence, including of course the Frobenius norm (<span class="math notranslate nohighlight">\(\beta=2\)</span>), the
(generalized) Kullback-Leibler divergence (<span class="math notranslate nohighlight">\(\beta=1\)</span>) and the
Itakura-Saito divergence (<span class="math notranslate nohighlight">\(\beta=0\)</span>). Note that for
<span class="math notranslate nohighlight">\(\beta \in (1; 2)\)</span>, the ‘mu’ solver is significantly faster than for other
values of <span class="math notranslate nohighlight">\(\beta\)</span>. Note also that with a negative (or 0, i.e.
‘itakura-saito’) <span class="math notranslate nohighlight">\(\beta\)</span>, the input matrix cannot contain zero values.</p>
<p>The ‘cd’ solver can only optimize the Frobenius norm. Due to the
underlying non-convexity of NMF, the different solvers may converge to
different minima, even when optimizing the same distance function.</p>
<p>NMF is best used with the <code class="docutils literal notranslate"><span class="pre">fit_transform</span></code> method, which returns the matrix W.
The matrix H is stored into the fitted model in the <code class="docutils literal notranslate"><span class="pre">components_</span></code> attribute;
the method <code class="docutils literal notranslate"><span class="pre">transform</span></code> will decompose a new matrix X_new based on these
stored components:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">],</span> <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <span class="n">NMF</span>
<span class="gp">>>> </span><span class="n">model</span> <span class="o">=</span> <span class="n">NMF</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">init</span><span class="o">=</span><span class="s1">'random'</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">W</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">H</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">components_</span>
<span class="gp">>>> </span><span class="n">X_new</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mf">6.1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mf">3.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">W_new</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_new</span><span class="p">)</span>
</pre></div>
</div>
<aside class="topic">
<p class="topic-title">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>
<li><p><a class="reference internal" href="../auto_examples/applications/plot_topics_extraction_with_nmf_lda.html#sphx-glr-auto-examples-applications-plot-topics-extraction-with-nmf-lda-py"><span class="std std-ref">Topic extraction with Non-negative Matrix Factorization and Latent Dirichlet Allocation</span></a></p></li>
</ul>
</aside>
</section>
<section id="mini-batch-non-negative-matrix-factorization">
<span id="minibatchnmf"></span><h3><span class="section-number">2.5.7.3. </span>Mini-batch Non Negative Matrix Factorization<a class="headerlink" href="#mini-batch-non-negative-matrix-factorization" title="Permalink to this heading">¶</a></h3>
<p><a class="reference internal" href="generated/sklearn.decomposition.MiniBatchNMF.html#sklearn.decomposition.MiniBatchNMF" title="sklearn.decomposition.MiniBatchNMF"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchNMF</span></code></a> <a class="footnote-reference brackets" href="#id18" id="id12" role="doc-noteref"><span class="fn-bracket">[</span>7<span class="fn-bracket">]</span></a> implements a faster, but less accurate version of the
non negative matrix factorization (i.e. <a class="reference internal" href="generated/sklearn.decomposition.NMF.html#sklearn.decomposition.NMF" title="sklearn.decomposition.NMF"><code class="xref py py-class docutils literal notranslate"><span class="pre">NMF</span></code></a>),
better suited for large datasets.</p>
<p>By default, <a class="reference internal" href="generated/sklearn.decomposition.MiniBatchNMF.html#sklearn.decomposition.MiniBatchNMF" title="sklearn.decomposition.MiniBatchNMF"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchNMF</span></code></a> divides the data into mini-batches and
optimizes the NMF model in an online manner by cycling over the mini-batches
for the specified number of iterations. The <code class="docutils literal notranslate"><span class="pre">batch_size</span></code> parameter controls
the size of the batches.</p>
<p>In order to speed up the mini-batch algorithm it is also possible to scale
past batches, giving them less importance than newer batches. This is done
introducing a so-called forgetting factor controlled by the <code class="docutils literal notranslate"><span class="pre">forget_factor</span></code>
parameter.</p>
<p>The estimator also implements <code class="docutils literal notranslate"><span class="pre">partial_fit</span></code>, which updates <code class="docutils literal notranslate"><span class="pre">H</span></code> by iterating
only once over a mini-batch. This can be used for online learning when the data
is not readily available from the start, or when the data does not fit into memory.</p>
<aside class="topic">
<p class="topic-title">References:</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id13" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id6">1</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="http://www.cs.columbia.edu/~blei/fogm/2020F/readings/LeeSeung1999.pdf">“Learning the parts of objects by non-negative matrix factorization”</a>
D. Lee, S. Seung, 1999</p>
</aside>
<aside class="footnote brackets" id="id14" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id7">2</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://www.jmlr.org/papers/volume5/hoyer04a/hoyer04a.pdf">“Non-negative Matrix Factorization with Sparseness Constraints”</a>
P. Hoyer, 2004</p>
</aside>
<aside class="footnote brackets" id="id15" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id8">4</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://www.boutsidis.org/Boutsidis_PRE_08.pdf">“SVD based initialization: A head start for nonnegative
matrix factorization”</a>
C. Boutsidis, E. Gallopoulos, 2008</p>
</aside>
<aside class="footnote brackets" id="id16" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id10">5</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://www.researchgate.net/profile/Anh-Huy-Phan/publication/220241471_Fast_Local_Algorithms_for_Large_Scale_Nonnegative_Matrix_and_Tensor_Factorizations">“Fast local algorithms for large scale nonnegative matrix and tensor
factorizations.”</a>
A. Cichocki, A. Phan, 2009</p>
</aside>
<aside class="footnote brackets" id="id17" role="note">
<span class="label"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id9">1</a>,<a role="doc-backlink" href="#id11">2</a>)</span>
<p><a class="reference external" href="https://arxiv.org/abs/1010.1763">“Algorithms for nonnegative matrix factorization with
the beta-divergence”</a>
C. Fevotte, J. Idier, 2011</p>
</aside>
<aside class="footnote brackets" id="id18" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id12">7</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://arxiv.org/abs/1106.4198">“Online algorithms for nonnegative matrix factorization with the
Itakura-Saito divergence”</a>
A. Lefevre, F. Bach, C. Fevotte, 2011</p>
</aside>
</aside>
</aside>
</section>
</section>
<section id="latent-dirichlet-allocation-lda">
<span id="latentdirichletallocation"></span><h2><span class="section-number">2.5.8. </span>Latent Dirichlet Allocation (LDA)<a class="headerlink" href="#latent-dirichlet-allocation-lda" title="Permalink to this heading">¶</a></h2>
<p>Latent Dirichlet Allocation is a generative probabilistic model for collections of
discrete dataset such as text corpora. It is also a topic model that is used for
discovering abstract topics from a collection of documents.</p>
<p>The graphical model of LDA is a three-level generative model:</p>
<img alt="../_images/lda_model_graph.png" class="align-center" src="../_images/lda_model_graph.png" />