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<ul>
<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="#kernel-pca">2.5.1.4. Kernel PCA</a></li>
<li><a class="reference internal" href="#sparse-principal-components-analysis-sparsepca-and-minibatchsparsepca">2.5.1.5. Sparse principal components analysis (SparsePCA and MiniBatchSparsePCA)</a></li>
</ul>
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<li><a class="reference internal" href="#truncated-singular-value-decomposition-and-latent-semantic-analysis">2.5.2. Truncated singular value decomposition and latent semantic analysis</a></li>
<li><a class="reference internal" href="#dictionary-learning">2.5.3. Dictionary Learning</a><ul>
<li><a class="reference internal" href="#sparse-coding-with-a-precomputed-dictionary">2.5.3.1. Sparse coding with a precomputed dictionary</a></li>
<li><a class="reference internal" href="#generic-dictionary-learning">2.5.3.2. Generic dictionary learning</a></li>
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<li><a class="reference internal" href="#independent-component-analysis-ica">2.5.5. Independent component analysis (ICA)</a></li>
<li><a class="reference internal" href="#non-negative-matrix-factorization-nmf-or-nnmf">2.5.6. Non-negative matrix factorization (NMF or NNMF)</a><ul>
<li><a class="reference internal" href="#nmf-with-the-frobenius-norm">2.5.6.1. NMF with the Frobenius norm</a></li>
<li><a class="reference internal" href="#nmf-with-a-beta-divergence">2.5.6.2. NMF with a beta-divergence</a></li>
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<div class="section" id="decomposing-signals-in-components-matrix-factorization-problems">
<span id="decompositions"></span><h1>2.5. Decomposing signals in components (matrix factorization problems)<a class="headerlink" href="#decomposing-signals-in-components-matrix-factorization-problems" title="Permalink to this headline">¶</a></h1>
<div class="section" id="principal-component-analysis-pca">
<span id="pca"></span><h2>2.5.1. Principal component analysis (PCA)<a class="headerlink" href="#principal-component-analysis-pca" title="Permalink to this headline">¶</a></h2>
<div class="section" id="exact-pca-and-probabilistic-interpretation">
<h3>2.5.1.1. Exact PCA and probabilistic interpretation<a class="headerlink" href="#exact-pca-and-probabilistic-interpretation" title="Permalink to this headline">¶</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"><span class="pre">PCA</span></code></a> is implemented as a <em>transformer</em> object
that learns <span class="math">\(n\)</span> components in its <code class="docutils literal"><span class="pre">fit</span></code> method, and can be used on new
data to project it on these components.</p>
<p>The optional parameter <code class="docutils literal"><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>
<div class="figure 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_0011.png" src="../_images/sphx_glr_plot_pca_vs_lda_0011.png" style="width: 480.0px; height: 360.0px;" /></a>
</div>
<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"><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
<cite>score</cite> method that can be used in cross-validation:</p>
<div class="figure 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_0011.png" src="../_images/sphx_glr_plot_pca_vs_fa_model_selection_0011.png" style="width: 480.0px; height: 360.0px;" /></a>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
<li><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></li>
</ul>
</div>
</div>
<div class="section" id="incremental-pca">
<span id="incrementalpca"></span><h3>2.5.1.2. Incremental PCA<a class="headerlink" href="#incremental-pca" title="Permalink to this headline">¶</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"><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"><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"><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"><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"><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>Using its <code class="docutils literal"><span class="pre">partial_fit</span></code> method on chunks of data fetched sequentially
from the local hard drive or a network database.</li>
<li>Calling its fit method on a memory mapped file using <code class="docutils literal"><span class="pre">numpy.memmap</span></code>.</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"><span class="pre">IncrementalPCA</span></code></a> only stores estimates of component and noise variances,
in order update <code class="docutils literal"><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>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_incremental_pca.html"><img alt="../_images/sphx_glr_plot_incremental_pca_0011.png" src="../_images/sphx_glr_plot_incremental_pca_0011.png" style="width: 600.0px; height: 600.0px;" /></a>
</div>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_incremental_pca.html"><img alt="../_images/sphx_glr_plot_incremental_pca_0021.png" src="../_images/sphx_glr_plot_incremental_pca_0021.png" style="width: 600.0px; height: 600.0px;" /></a>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
</div>
<div class="section" id="pca-using-randomized-svd">
<span id="randomizedpca"></span><h3>2.5.1.3. PCA using randomized SVD<a class="headerlink" href="#pca-using-randomized-svd" title="Permalink to this headline">¶</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"><span class="pre">PCA</span></code></a> used with the optional parameter
<code class="docutils literal"><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">\(n_{samples} = 400\)</span>
and <span class="math">\(n_{features} = 64 \times 64 = 4096\)</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_0011.png" style="width: 360.0px; height: 270.59999999999997px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="pca_img" src="../_images/sphx_glr_plot_faces_decomposition_0021.png" style="width: 360.0px; height: 270.59999999999997px;" /></a></strong></p><p>If we note <span class="math">\(n_{\max} = \max(n_{\mathrm{samples}}, n_{\mathrm{features}})\)</span> and
<span class="math">\(n_{\min} = \min(n_{\mathrm{samples}}, n_{\mathrm{features}})\)</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"><span class="pre">PCA</span></code></a> is <span class="math">\(O(n_{\max}^2 \cdot n_{\mathrm{components}})\)</span>
instead of <span class="math">\(O(n_{\max}^2 \cdot n_{\min})\)</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"><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"><span class="pre">PCA</span></code></a> is also proportional to
<span class="math">\(2 \cdot n_{\max} \cdot n_{\mathrm{components}}\)</span> instead of <span class="math">\(n_{\max}
\cdot n_{\min}\)</span> for the exact method.</p>
<p>Note: the implementation of <code class="docutils literal"><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"><span class="pre">PCA</span></code></a> with
<code class="docutils literal"><span class="pre">svd_solver='randomized'</span></code> is not the exact inverse transform of
<code class="docutils literal"><span class="pre">transform</span></code> even when <code class="docutils literal"><span class="pre">whiten=False</span></code> (default).</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
<li><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></li>
</ul>
</div>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li><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</li>
</ul>
</div>
</div>
<div class="section" id="kernel-pca">
<span id="id1"></span><h3>2.5.1.4. Kernel PCA<a class="headerlink" href="#kernel-pca" title="Permalink to this headline">¶</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"><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>). 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"><span class="pre">KernelPCA</span></code></a> supports both
<code class="docutils literal"><span class="pre">transform</span></code> and <code class="docutils literal"><span class="pre">inverse_transform</span></code>.</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_kernel_pca.html"><img alt="../_images/sphx_glr_plot_kernel_pca_0011.png" src="../_images/sphx_glr_plot_kernel_pca_0011.png" style="width: 480.0px; height: 360.0px;" /></a>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
</div>
<div class="section" id="sparse-principal-components-analysis-sparsepca-and-minibatchsparsepca">
<span id="sparsepca"></span><h3>2.5.1.5. Sparse principal components analysis (SparsePCA and MiniBatchSparsePCA)<a class="headerlink" href="#sparse-principal-components-analysis-sparsepca-and-minibatchsparsepca" title="Permalink to this headline">¶</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"><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"><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"><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"><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">\(h \in \mathbf{R}^{4096}\)</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="id2">[Jen09]</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_0021.png" style="width: 360.0px; height: 270.59999999999997px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="spca_img" src="../_images/sphx_glr_plot_faces_decomposition_0051.png" style="width: 360.0px; height: 270.59999999999997px;" /></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="id3">[Mrl09]</a> . The optimization
problem solved is a PCA problem (dictionary learning) with an
<span class="math">\(\ell_1\)</span> penalty on the components:</p>
<div class="math">
\[\begin{split}(U^*, V^*) = \underset{U, V}{\operatorname{arg\,min\,}} & \frac{1}{2}
||X-UV||_2^2+\alpha||V||_1 \\
\text{subject to } & ||U_k||_2 = 1 \text{ for all }
0 \leq k < n_{components}\end{split}\]</div>
<p>The sparsity-inducing <span class="math">\(\ell_1\)</span> 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"><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="first admonition-title">Note</p>
<p class="last">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"><span class="pre">MiniBatchSparsePCA</span></code></a> does not implement <code class="docutils literal"><span class="pre">partial_fit</span></code> because
the algorithm is online along the features direction, not the samples
direction.</p>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
<div class="topic">
<p class="topic-title first">References:</p>
<table class="docutils citation" frame="void" id="mrl09" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id3">[Mrl09]</a></td><td><a class="reference external" href="http://www.di.ens.fr/sierra/pdfs/icml09.pdf">“Online Dictionary Learning for Sparse Coding”</a>
J. Mairal, F. Bach, J. Ponce, G. Sapiro, 2009</td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="jen09" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id2">[Jen09]</a></td><td><a class="reference external" href="www.di.ens.fr/~fbach/sspca_AISTATS2010.pdf">“Structured Sparse Principal Component Analysis”</a>
R. Jenatton, G. Obozinski, F. Bach, 2009</td></tr>
</tbody>
</table>
</div>
</div>
</div>
<div class="section" id="truncated-singular-value-decomposition-and-latent-semantic-analysis">
<span id="lsa"></span><h2>2.5.2. Truncated singular value decomposition and latent semantic analysis<a class="headerlink" href="#truncated-singular-value-decomposition-and-latent-semantic-analysis" title="Permalink to this headline">¶</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"><span class="pre">TruncatedSVD</span></code></a> implements a variant of singular value decomposition
(SVD) that only computes the <span class="math">\(k\)</span> largest singular values,
where <span class="math">\(k\)</span> is a user-specified parameter.</p>
<p>When truncated SVD is applied to term-document matrices
(as returned by <code class="docutils literal"><span class="pre">CountVectorizer</span></code> or <code class="docutils literal"><span class="pre">TfidfVectorizer</span></code>),
this transformation is known as
<a class="reference external" href="http://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="first admonition-title">Note</p>
<p class="last">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">\(X\)</span>
produces a low-rank approximation <span class="math">\(X\)</span>:</p>
<div class="math">
\[X \approx X_k = U_k \Sigma_k V_k^\top\]</div>
<p>After this operation, <span class="math">\(U_k \Sigma_k^\top\)</span>
is the transformed training set with <span class="math">\(k\)</span> features
(called <code class="docutils literal"><span class="pre">n_components</span></code> in the API).</p>
<p>To also transform a test set <span class="math">\(X\)</span>, we multiply it with <span class="math">\(V_k\)</span>:</p>
<div class="math">
\[X' = X V_k\]</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">Most treatments of LSA in the natural language processing (NLP)
and information retrieval (IR) literature
swap the axes of the matrix <span class="math">\(X\)</span> so that it has shape
<code class="docutils literal"><span class="pre">n_features</span></code> × <code class="docutils literal"><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"><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"><span class="pre">PCA</span></code></a>, but differs
in that it works on sample matrices <span class="math">\(X\)</span> directly
instead of their covariance matrices.
When the columnwise (per-feature) means of <span class="math">\(X\)</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"><span class="pre">TruncatedSVD</span></code></a> transformer accepts <code class="docutils literal"><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"><span class="pre">TruncatedSVD</span></code></a> transformer
works with any (sparse) 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"><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>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li>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="http://nlp.stanford.edu/IR-book/pdf/18lsi.pdf">Matrix decompositions & latent semantic indexing</a></li>
</ul>
</div>
</div>
<div class="section" id="dictionary-learning">
<span id="dictionarylearning"></span><h2>2.5.3. Dictionary Learning<a class="headerlink" href="#dictionary-learning" title="Permalink to this headline">¶</a></h2>
<div class="section" id="sparse-coding-with-a-precomputed-dictionary">
<span id="sparsecoder"></span><h3>2.5.3.1. Sparse coding with a precomputed dictionary<a class="headerlink" href="#sparse-coding-with-a-precomputed-dictionary" title="Permalink to this headline">¶</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"><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"><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"><span class="pre">transform_method</span></code> initialization parameter:</p>
<ul class="simple">
<li>Orthogonal matching pursuit (<a class="reference internal" href="linear_model.html#omp"><span class="std std-ref">Orthogonal Matching Pursuit (OMP)</span></a>)</li>
<li>Least-angle regression (<a class="reference internal" href="linear_model.html#least-angle-regression"><span class="std std-ref">Least Angle Regression</span></a>)</li>
<li>Lasso computed by least-angle regression</li>
<li>Lasso using coordinate descent (<a class="reference internal" href="linear_model.html#lasso"><span class="std std-ref">Lasso</span></a>)</li>
<li>Thresholding</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"><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"><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"><span class="pre">n_components</span></code> is computed. Then, the first <code class="docutils literal"><span class="pre">n_components</span></code> entries of the
<code class="docutils literal"><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>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
</div>
<div class="section" id="generic-dictionary-learning">
<h3>2.5.3.2. Generic dictionary learning<a class="headerlink" href="#generic-dictionary-learning" title="Permalink to this headline">¶</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"><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">
\[\begin{split}(U^*, V^*) = \underset{U, V}{\operatorname{arg\,min\,}} & \frac{1}{2}
||X-UV||_2^2+\alpha||U||_1 \\
\text{subject to } & ||V_k||_2 = 1 \text{ for all }
0 \leq k < n_{\mathrm{atoms}}\end{split}\]</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_0021.png" style="width: 360.0px; height: 270.59999999999997px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="dict_img2" src="../_images/sphx_glr_plot_faces_decomposition_0061.png" style="width: 360.0px; height: 270.59999999999997px;" /></a></strong></p><p>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_0111.png" style="width: 360.0px; height: 270.59999999999997px;" /></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_0121.png" style="width: 360.0px; height: 270.59999999999997px;" /></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_0131.png" style="width: 360.0px; height: 270.59999999999997px;" /></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_0141.png" style="width: 360.0px; height: 270.59999999999997px;" /></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>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_image_denoising.html"><img alt="../_images/sphx_glr_plot_image_denoising_0011.png" src="../_images/sphx_glr_plot_image_denoising_0011.png" style="width: 210.0px; height: 200.0px;" /></a>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li><a class="reference external" href="http://www.di.ens.fr/sierra/pdfs/icml09.pdf">“Online dictionary learning for sparse coding”</a>
J. Mairal, F. Bach, J. Ponce, G. Sapiro, 2009</li>
</ul>
</div>
</div>
<div class="section" id="mini-batch-dictionary-learning">
<span id="minibatchdictionarylearning"></span><h3>2.5.3.3. Mini-batch dictionary learning<a class="headerlink" href="#mini-batch-dictionary-learning" title="Permalink to this headline">¶</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"><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"><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"><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_0011.png" class="align-right" src="../_images/sphx_glr_plot_dict_face_patches_0011.png" style="width: 210.0px; height: 200.0px;" /></a>
<div class="topic">
<p class="topic-title first"><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"><span class="pre">MiniBatchKMeans</span></code></a> estimator is
computationally efficient and implements on-line learning with a
<code class="docutils literal"><span class="pre">partial_fit</span></code> method.</p>
<blockquote>
<div>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></div></blockquote>
</div>
</div>
</div>
<div class="section" id="factor-analysis">
<span id="fa"></span><h2>2.5.4. Factor Analysis<a class="headerlink" href="#factor-analysis" title="Permalink to this headline">¶</a></h2>
<p>In unsupervised learning we only have a dataset <span class="math">\(X = \{x_1, x_2, \dots, x_n
\}\)</span>. How can this dataset be described mathematically? A very simple
<cite>continuous latent variable</cite> model for <span class="math">\(X\)</span> is</p>
<div class="math">
\[x_i = W h_i + \mu + \epsilon\]</div>
<p>The vector <span class="math">\(h_i\)</span> is called “latent” because it is unobserved. <span class="math">\(\epsilon\)</span> is
considered a noise term distributed according to a Gaussian with mean 0 and
covariance <span class="math">\(\Psi\)</span> (i.e. <span class="math">\(\epsilon \sim \mathcal{N}(0, \Psi)\)</span>), <span class="math">\(\mu\)</span> is some
arbitrary offset vector. Such a model is called “generative” as it describes
how <span class="math">\(x_i\)</span> is generated from <span class="math">\(h_i\)</span>. If we use all the <span class="math">\(x_i\)</span>‘s as columns to form
a matrix <span class="math">\(\mathbf{X}\)</span> and all the <span class="math">\(h_i\)</span>‘s as columns of a matrix <span class="math">\(\mathbf{H}\)</span>
then we can write (with suitably defined <span class="math">\(\mathbf{M}\)</span> and <span class="math">\(\mathbf{E}\)</span>):</p>
<div class="math">
\[\mathbf{X} = W \mathbf{H} + \mathbf{M} + \mathbf{E}\]</div>
<p>In other words, we <em>decomposed</em> matrix <span class="math">\(\mathbf{X}\)</span>.</p>
<p>If <span class="math">\(h_i\)</span> is given, the above equation automatically implies the following
probabilistic interpretation:</p>
<div class="math">
\[p(x_i|h_i) = \mathcal{N}(Wh_i + \mu, \Psi)\]</div>
<p>For a complete probabilistic model we also need a prior distribution for the
latent variable <span class="math">\(h\)</span>. The most straightforward assumption (based on the nice
properties of the Gaussian distribution) is <span class="math">\(h \sim \mathcal{N}(0,
\mathbf{I})\)</span>. This yields a Gaussian as the marginal distribution of <span class="math">\(x\)</span>:</p>
<div class="math">
\[p(x) = \mathcal{N}(\mu, WW^T + \Psi)\]</div>
<p>Now, without any further assumptions the idea of having a latent variable <span class="math">\(h\)</span>
would be superfluous – <span class="math">\(x\)</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">\(\Psi\)</span>:</p>
<ul class="simple">
<li><span class="math">\(\Psi = \sigma^2 \mathbf{I}\)</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"><span class="pre">PCA</span></code></a>.</li>
<li><span class="math">\(\Psi = \mathrm{diag}(\psi_1, \psi_2, \dots, \psi_n)\)</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"><span class="pre">FactorAnalysis</span></code></a>, a classical statistical model. The matrix W is
sometimes called the “factor loading matrix”.</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"><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"><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_0021.png" style="width: 360.0px; height: 270.59999999999997px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="fa_img3" src="../_images/sphx_glr_plot_faces_decomposition_0091.png" style="width: 360.0px; height: 270.59999999999997px;" /></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"><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>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="../_images/sphx_glr_plot_faces_decomposition_0081.png" src="../_images/sphx_glr_plot_faces_decomposition_0081.png" style="width: 150.0px; height: 168.75px;" /></a>
</div>
<p>This allows better model selection than probabilistic PCA in the presence
of heteroscedastic noise:</p>
<div class="figure 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_0021.png" src="../_images/sphx_glr_plot_pca_vs_fa_model_selection_0021.png" style="width: 480.0px; height: 360.0px;" /></a>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
</div>
<div class="section" id="independent-component-analysis-ica">
<span id="ica"></span><h2>2.5.5. Independent component analysis (ICA)<a class="headerlink" href="#independent-component-analysis-ica" title="Permalink to this headline">¶</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"><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>
<div class="figure 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_0011.png" src="../_images/sphx_glr_plot_ica_blind_source_separation_0011.png" style="width: 384.0px; height: 288.0px;" /></a>
</div>
<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_0021.png" style="width: 360.0px; height: 270.59999999999997px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="ica_img4" src="../_images/sphx_glr_plot_faces_decomposition_0041.png" style="width: 360.0px; height: 270.59999999999997px;" /></a></strong></p><div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
<li><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></li>
<li><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></li>
</ul>
</div>
</div>
<div class="section" id="non-negative-matrix-factorization-nmf-or-nnmf">
<span id="nmf"></span><h2>2.5.6. Non-negative matrix factorization (NMF or NNMF)<a class="headerlink" href="#non-negative-matrix-factorization-nmf-or-nnmf" title="Permalink to this headline">¶</a></h2>
<div class="section" id="nmf-with-the-frobenius-norm">
<h3>2.5.6.1. NMF with the Frobenius norm<a class="headerlink" href="#nmf-with-the-frobenius-norm" title="Permalink to this headline">¶</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"><span class="pre">NMF</span></code></a> <a class="footnote-reference" href="#id11" id="id5">[1]</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"><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"><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">\(X\)</span> into two matrices <span class="math">\(W\)</span> and <span class="math">\(H\)</span> of non-negative elements,
by optimizing the distance <span class="math">\(d\)</span> between <span class="math">\(X\)</span> and the matrix product
<span class="math">\(WH\)</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">
\[d_{\mathrm{Fro}}(X, Y) = \frac{1}{2} ||X - Y||_{\mathrm{Fro}}^2 = \frac{1}{2} \sum_{i,j} (X_{ij} - {Y}_{ij})^2\]</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"><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" href="#id12" id="id6">[2]</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"><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"><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_0021.png" style="width: 360.0px; height: 270.59999999999997px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="nmf_img5" src="../_images/sphx_glr_plot_faces_decomposition_0031.png" style="width: 360.0px; height: 270.59999999999997px;" /></a></strong></p><p>The <code class="xref py py-attr docutils literal"><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"><span class="pre">NMF</span></code></a> implements the
method Nonnegative Double Singular Value Decomposition. NNDSVD <a class="footnote-reference" href="#id13" id="id7">[4]</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"><span class="pre">NMF</span></code></a> can also be initialized with correctly scaled random non-negative
matrices by setting <code class="xref py py-attr docutils literal"><span class="pre">init="random"</span></code>. An integer seed or a
<code class="docutils literal"><span class="pre">RandomState</span></code> can also be passed to <code class="xref py py-attr docutils literal"><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"><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 <code class="xref py py-class docutils literal"><span class="pre">ElasticNet</span></code>, we control the
combination of L1 and L2 with the <code class="xref py py-attr docutils literal"><span class="pre">l1_ratio</span></code> (<span class="math">\(\rho\)</span>) parameter,
and the intensity of the regularization with the <code class="xref py py-attr docutils literal"><span class="pre">alpha</span></code>
(<span class="math">\(\alpha\)</span>) parameter. Then the priors terms are:</p>
<div class="math">
\[\alpha \rho ||W||_1 + \alpha \rho ||H||_1
+ \frac{\alpha(1-\rho)}{2} ||W||_{\mathrm{Fro}} ^ 2
+ \frac{\alpha(1-\rho)}{2} ||H||_{\mathrm{Fro}} ^ 2\]</div>
<p>and the regularized objective function is:</p>
<div class="math">
\[d_{\mathrm{Fro}}(X, WH)
+ \alpha \rho ||W||_1 + \alpha \rho ||H||_1
+ \frac{\alpha(1-\rho)}{2} ||W||_{\mathrm{Fro}} ^ 2
+ \frac{\alpha(1-\rho)}{2} ||H||_{\mathrm{Fro}} ^ 2\]</div>
<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"><span class="pre">NMF</span></code></a> regularizes both W and H. The public function
<code class="xref py py-func docutils literal"><span class="pre">non_negative_factorization</span></code> allows a finer control through the
<code class="xref py py-attr docutils literal"><span class="pre">regularization</span></code> attribute, and may regularize only W, only H, or both.</p>
</div>
<div class="section" id="nmf-with-a-beta-divergence">
<h3>2.5.6.2. NMF with a beta-divergence<a class="headerlink" href="#nmf-with-a-beta-divergence" title="Permalink to this headline">¶</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">
\[d_{\mathrm{Fro}}(X, Y) = \frac{1}{2} ||X - Y||_{Fro}^2 = \frac{1}{2} \sum_{i,j} (X_{ij} - {Y}_{ij})^2\]</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">
\[d_{KL}(X, Y) = \sum_{i,j} (X_{ij} \log(\frac{X_{ij}}{Y_{ij}}) - X_{ij} + Y_{ij})\]</div>
<p>Or, the Itakura-Saito (IS) divergence:</p>
<div class="math">
\[d_{IS}(X, Y) = \sum_{i,j} (\frac{X_{ij}}{Y_{ij}} - \log(\frac{X_{ij}}{Y_{ij}}) - 1)\]</div>
<p>These three distances are special cases of the beta-divergence family, with
<span class="math">\(\beta = 2, 1, 0\)</span> respectively <a class="footnote-reference" href="#id15" id="id8">[6]</a>. The beta-divergence are
defined by :</p>
<div class="math">
\[d_{\beta}(X, Y) = \sum_{i,j} \frac{1}{\beta(\beta - 1)}(X_{ij}^\beta + (\beta-1)Y_{ij}^\beta - \beta X_{ij} Y_{ij}^{\beta - 1})\]</div>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_beta_divergence.html"><img alt="../_images/sphx_glr_plot_beta_divergence_0011.png" src="../_images/sphx_glr_plot_beta_divergence_0011.png" style="width: 480.0px; height: 360.0px;" /></a>
</div>
<p>Note that this definition is not valid if <span class="math">\(\beta \in (0; 1)\)</span>, yet it can
be continuously extended to the definitions of <span class="math">\(d_{KL}\)</span> and <span class="math">\(d_{IS}\)</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"><span class="pre">NMF</span></code></a> implements two solvers, using Coordinate Descent (‘cd’) <a class="footnote-reference" href="#id14" id="id9">[5]</a>, and
Multiplicative Update (‘mu’) <a class="footnote-reference" href="#id15" id="id10">[6]</a>. The ‘mu’ solver can optimize every
beta-divergence, including of course the Frobenius norm (<span class="math">\(\beta=2\)</span>), the
(generalized) Kullback-Leibler divergence (<span class="math">\(\beta=1\)</span>) and the
Itakura-Saito divergence (<span class="math">\(\beta=0\)</span>). Note that for
<span class="math">\(\beta \in (1; 2)\)</span>, the ‘mu’ solver is significantly faster than for other
values of <span class="math">\(\beta\)</span>. Note also that with a negative (or 0, i.e.
‘itakura-saito’) <span class="math">\(\beta\)</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"><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"><span class="pre">components_</span></code> attribute;
the method <code class="docutils literal"><span class="pre">transform</span></code> will decompose a new matrix X_new based on these
stored components:</p>
<div class="highlight-default"><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="k">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>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
<li><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></li>
<li><a class="reference internal" href="../auto_examples/decomposition/plot_beta_divergence.html#sphx-glr-auto-examples-decomposition-plot-beta-divergence-py"><span class="std std-ref">Beta-divergence loss functions</span></a></li>
</ul>
</div>
<div class="topic">
<p class="topic-title first">References:</p>
<table class="docutils footnote" frame="void" id="id11" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id5">[1]</a></td><td><a class="reference external" href="http://www.columbia.edu/~jwp2128/Teaching/E4903/papers/nmf_nature.pdf">“Learning the parts of objects by non-negative matrix factorization”</a>
D. Lee, S. Seung, 1999</td></tr>
</tbody>
</table>
<table class="docutils footnote" frame="void" id="id12" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id6">[2]</a></td><td><a class="reference external" href="http://www.jmlr.org/papers/volume5/hoyer04a/hoyer04a.pdf">“Non-negative Matrix Factorization with Sparseness Constraints”</a>
P. Hoyer, 2004</td></tr>
</tbody>
</table>
<table class="docutils footnote" frame="void" id="id13" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id7">[4]</a></td><td><a class="reference external" href="http://scgroup.hpclab.ceid.upatras.gr/faculty/stratis/Papers/HPCLAB020107.pdf">“SVD based initialization: A head start for nonnegative
matrix factorization”</a>
C. Boutsidis, E. Gallopoulos, 2008</td></tr>
</tbody>
</table>
<table class="docutils footnote" frame="void" id="id14" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id9">[5]</a></td><td><a class="reference external" href="http://www.bsp.brain.riken.jp/publications/2009/Cichocki-Phan-IEICE_col.pdf">“Fast local algorithms for large scale nonnegative matrix and tensor
factorizations.”</a>
A. Cichocki, A. Phan, 2009</td></tr>
</tbody>
</table>
<table class="docutils footnote" frame="void" id="id15" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label">[6]</td><td><em>(<a class="fn-backref" href="#id8">1</a>, <a class="fn-backref" href="#id10">2</a>)</em> <a class="reference external" href="https://arxiv.org/pdf/1010.1763.pdf">“Algorithms for nonnegative matrix factorization with the beta-divergence”</a>
C. Fevotte, J. Idier, 2011</td></tr>
</tbody>
</table>
</div>
</div>
</div>
<div class="section" id="latent-dirichlet-allocation-lda">
<span id="latentdirichletallocation"></span><h2>2.5.7. Latent Dirichlet Allocation (LDA)<a class="headerlink" href="#latent-dirichlet-allocation-lda" title="Permalink to this headline">¶</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" />
<p>Note on notations presented in the graphical model above, which can be found in
Hoffman et al. (2013):</p>
<blockquote>
<div><ul class="simple">
<li>The corpus is a collection of <span class="math">\(D\)</span> documents.</li>
<li>A document is a sequence of <span class="math">\(N\)</span> words.</li>
<li>There are <span class="math">\(K\)</span> topics in the corpus.</li>
<li>The boxes represent repeated sampling.</li>
</ul>
</div></blockquote>
<p>In the graphical model, each node is a random variable and has a role in the
generative process. A shaded node indicates an observed variable and an unshaded
node indicates a hidden (latent) variable. In this case, words in the corpus are
the only data that we observe. The latent variables determine the random mixture
of topics in the corpus and the distribution of words in the documents.
The goal of LDA is to use the observed words to infer the hidden topic
structure.</p>
<p>When modeling text corpora, the model assumes the following generative process
for a corpus with <span class="math">\(D\)</span> documents and <span class="math">\(K\)</span> topics, with <span class="math">\(K\)</span>
corresponding to <code class="xref py py-attr docutils literal"><span class="pre">n_components</span></code> in the API:</p>
<blockquote>
<div><ol class="arabic simple">
<li>For each topic <span class="math">\(k \in K\)</span>, draw <span class="math">\(\beta_k \sim
\mathrm{Dirichlet}(\eta)\)</span>. This provides a distribution over the words,
i.e. the probability of a word appearing in topic <span class="math">\(k\)</span>.
<span class="math">\(\eta\)</span> corresponds to <code class="xref py py-attr docutils literal"><span class="pre">topic_word_prior</span></code>.</li>
<li>For each document <span class="math">\(d \in D\)</span>, draw the topic proportions
<span class="math">\(\theta_d \sim \mathrm{Dirichlet}(\alpha)\)</span>. <span class="math">\(\alpha\)</span>
corresponds to <code class="xref py py-attr docutils literal"><span class="pre">doc_topic_prior</span></code>.</li>
<li>For each word <span class="math">\(i\)</span> in document <span class="math">\(d\)</span>:</li>
</ol>
<blockquote>
<div><ol class="loweralpha simple">
<li>Draw the topic assignment <span class="math">\(z_{di} \sim \mathrm{Multinomial}
(\theta_d)\)</span></li>
<li>Draw the observed word <span class="math">\(w_{ij} \sim \mathrm{Multinomial}
(\beta_{z_{di}})\)</span></li>
</ol>
</div></blockquote>
</div></blockquote>
<p>For parameter estimation, the posterior distribution is:</p>
<div class="math">
\[p(z, \theta, \beta |w, \alpha, \eta) =
\frac{p(z, \theta, \beta|\alpha, \eta)}{p(w|\alpha, \eta)}\]</div>
<p>Since the posterior is intractable, variational Bayesian method
uses a simpler distribution <span class="math">\(q(z,\theta,\beta | \lambda, \phi, \gamma)\)</span>
to approximate it, and those variational parameters <span class="math">\(\lambda\)</span>,
<span class="math">\(\phi\)</span>, <span class="math">\(\gamma\)</span> are optimized to maximize the Evidence
Lower Bound (ELBO):</p>
<div class="math">
\[\log\: P(w | \alpha, \eta) \geq L(w,\phi,\gamma,\lambda) \overset{\triangle}{=}
E_{q}[\log\:p(w,z,\theta,\beta|\alpha,\eta)] - E_{q}[\log\:q(z, \theta, \beta)]\]</div>
<p>Maximizing ELBO is equivalent to minimizing the Kullback-Leibler(KL) divergence
between <span class="math">\(q(z,\theta,\beta)\)</span> and the true posterior
<span class="math">\(p(z, \theta, \beta |w, \alpha, \eta)\)</span>.</p>
<p><a class="reference internal" href="generated/sklearn.decomposition.LatentDirichletAllocation.html#sklearn.decomposition.LatentDirichletAllocation" title="sklearn.decomposition.LatentDirichletAllocation"><code class="xref py py-class docutils literal"><span class="pre">LatentDirichletAllocation</span></code></a> implements the online variational Bayes
algorithm and supports both online and batch update methods.
While the batch method updates variational variables after each full pass through
the data, the online method updates variational variables from mini-batch data
points.</p>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">Although the online method is guaranteed to converge to a local optimum point, the quality of
the optimum point and the speed of convergence may depend on mini-batch size and
attributes related to learning rate setting.</p>
</div>
<p>When <a class="reference internal" href="generated/sklearn.decomposition.LatentDirichletAllocation.html#sklearn.decomposition.LatentDirichletAllocation" title="sklearn.decomposition.LatentDirichletAllocation"><code class="xref py py-class docutils literal"><span class="pre">LatentDirichletAllocation</span></code></a> is applied on a “document-term” matrix, the matrix
will be decomposed into a “topic-term” matrix and a “document-topic” matrix. While
“topic-term” matrix is stored as <code class="xref py py-attr docutils literal"><span class="pre">components_</span></code> in the model, “document-topic” matrix
can be calculated from <code class="docutils literal"><span class="pre">transform</span></code> method.</p>
<p><a class="reference internal" href="generated/sklearn.decomposition.LatentDirichletAllocation.html#sklearn.decomposition.LatentDirichletAllocation" title="sklearn.decomposition.LatentDirichletAllocation"><code class="xref py py-class docutils literal"><span class="pre">LatentDirichletAllocation</span></code></a> also implements <code class="docutils literal"><span class="pre">partial_fit</span></code> method. This is used
when data can be fetched sequentially.</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li><a class="reference external" href="http://www.jmlr.org/papers/volume3/blei03a/blei03a.pdf">“Latent Dirichlet Allocation”</a>
D. Blei, A. Ng, M. Jordan, 2003</li>
<li><a class="reference external" href="https://papers.nips.cc/paper/3902-online-learning-for-latent-dirichlet-allocation.pdf">“Online Learning for Latent Dirichlet Allocation”</a>
M. Hoffman, D. Blei, F. Bach, 2010</li>
<li><a class="reference external" href="http://www.columbia.edu/~jwp2128/Papers/HoffmanBleiWangPaisley2013.pdf">“Stochastic Variational Inference”</a>
M. Hoffman, D. Blei, C. Wang, J. Paisley, 2013</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
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