scikit-learn Decomposing signals in components matrix factorization problems sklearn decomposition decompositions

Answer:
.. _decompositions:

=================================================================
Decomposing signals in components (matrix factorization problems)
=================================================================

.. currentmodule:: sklearn.decomposition

.. _PCA:

Principal component analysis (PCA)
==================================

Exact PCA and probabilistic interpretation
------------------------------------------

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, :class:`PCA` is implemented as a *transformer* object
that learns :math:`n` components in its ``fit`` method, and can be used on new
data to project it on these components.

PCA centers but does not scale the input data for each feature before
applying the SVD. The optional parameter ``whiten=True`` 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.

Below is an example of the iris dataset, which is comprised of 4
features, projected on the 2 dimensions that explain most variance:

.. figure:: ../auto_examples/decomposition/images/sphx_glr_plot_pca_vs_lda_001.png
    :target: ../auto_examples/decomposition/plot_pca_vs_lda.html
    :align: center
    :scale: 75%

The :class:`PCA` 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
:term:`score` method that can be used in cross-validation:

.. figure:: ../auto_examples/decomposition/images/sphx_glr_plot_pca_vs_fa_model_selection_001.png
    :target: ../auto_examples/decomposition/plot_pca_vs_fa_model_selection.html
    :align: center
    :scale: 75%

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_decomposition_plot_pca_iris.py`
* :ref:`sphx_glr_auto_examples_decomposition_plot_pca_vs_lda.py`
* :ref:`sphx_glr_auto_examples_decomposition_plot_pca_vs_fa_model_selection.py`

.. _IncrementalPCA:

Incremental PCA
---------------

The :class:`PCA` object is very useful, but has certain limitations for
large datasets. The biggest limitation is that :class:`PCA` only supports
batch processing, which means all of the data to be processed must fit in main
memory. The :class:`IncrementalPCA` object uses a different form of
processing and allows for partial computations which almost
exactly match the results of :class:`PCA` while processing the data in a
minibatch fashion. :class:`IncrementalPCA` makes it possible to implement
out-of-core Principal Component Analysis either by:

* Using its ``partial_fit`` method on chunks of data fetched sequentially
  from the local hard drive or a network database.

* Calling its fit method on a memory mapped file using
  ``numpy.memmap``.

:class:`IncrementalPCA` only stores estimates of component and noise variances,
in order update ``explained_variance_ratio_`` 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.

As in :class:`PCA`, :class:`IncrementalPCA` centers but does not scale the
input data for each feature before applying the SVD.

.. figure:: ../auto_examples/decomposition/images/sphx_glr_plot_incremental_pca_001.png
    :target: ../auto_examples/decomposition/plot_incremental_pca.html
    :align: center
    :scale: 75%

.. figure:: ../auto_examples/decomposition/images/sphx_glr_plot_incremental_pca_002.png
    :target: ../auto_examples/decomposition/plot_incremental_pca.html
    :align: center
    :scale: 75%

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_decomposition_plot_incremental_pca.py`

.. _RandomizedPCA:

PCA using randomized SVD
------------------------

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.

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.

The class :class:`PCA` used with the optional parameter
``svd_solver='randomized'`` 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.

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 :math:`n_{samples} = 400`
and :math:`n_{features} = 64 \times 64 = 4096`, the computation time is
less than 1s:

.. |orig_img| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_001.png
   :target: ../auto_examples/decomposition/plot_faces_decomposition.html
   :scale: 60%

.. |pca_img| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_002.png
   :target: ../auto_examples/decomposition/plot_faces_decomposition.html
   :scale: 60%

.. centered:: |orig_img| |pca_img|

If we note :math:`n_{\max} = \max(n_{\mathrm{samples}}, n_{\mathrm{features}})` and
:math:`n_{\min} = \min(n_{\mathrm{samples}}, n_{\mathrm{features}})`, the time complexity
of the randomized :class:`PCA` is :math:`O(n_{\max}^2 \cdot n_{\mathrm{components}})`
instead of :math:`O(n_{\max}^2 \cdot n_{\min})` for the exact method
implemented in :class:`PCA`.

The memory footprint of randomized :class:`PCA` is also proportional to
:math:`2 \cdot n_{\max} \cdot n_{\mathrm{components}}` instead of :math:`n_{\max}
\cdot n_{\min}` for the exact method.

Note: the implementation of ``inverse_transform`` in :class:`PCA` with
``svd_solver='randomized'`` is not the exact inverse transform of
``transform`` even when ``whiten=False`` (default).

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_applications_plot_face_recognition.py`
* :ref:`sphx_glr_auto_examples_decomposition_plot_faces_decomposition.py`

.. rubric:: References

* Algorithm 4.3 in
  :arxiv:`"Finding structure with randomness: Stochastic algorithms for
  constructing approximate matrix decompositions" <0909.4061>`
  Halko, et al., 2009

* :arxiv:`"An implementation of a randomized algorithm for principal component
  analysis" <1412.3510>` A. Szlam et al. 2014

.. _SparsePCA:

Sparse principal components analysis (SparsePCA and MiniBatchSparsePCA)
-----------------------------------------------------------------------

:class:`SparsePCA` is a variant of PCA, with the goal of extracting the
set of sparse components that best reconstruct the data.

Mini-batch sparse PCA (:class:`MiniBatchSparsePCA`) is a variant of
:class:`SparsePCA` 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.

Principal component analysis (:class:`PCA`) 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.

Sparse principal components yields a more parsimonious, interpretable
representation, clearly emphasizing which of the original features contribute
to the differences between samples.

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 :math:`h \in \mathbf{R}^{4096}`, 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
[Jen09]_ for a review of such methods.
For more details on how to use Sparse PCA, see the Examples section, below.

.. |spca_img| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_005.png
   :target: ../auto_examples/decomposition/plot_faces_decomposition.html
   :scale: 60%

.. centered:: |pca_img| |spca_img|

Note that there are many different formulations for the Sparse PCA
problem. The one implemented here is based on [Mrl09]_ . The optimization
problem solved is a PCA problem (dictionary learning) with an
:math:`\ell_1` penalty on the components:

.. math::
   (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}

:math:`||.||_{\text{Fro}}` stands for the Frobenius norm and :math:`||.||_{1,1}`
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 :math:`||.||_{1,1}` 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 ``alpha``. Small values lead to a gently regularized
factorization, while larger values shrink many coefficients to zero.

.. note::

  While in the spirit of an online algorithm, the class
  :class:`MiniBatchSparsePCA` does not implement ``partial_fit`` because
  the algorithm is online along the features direction, not the samples
  direction.

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_decomposition_plot_faces_decomposition.py`

.. rubric:: References

.. [Mrl09] `"Online Dictionary Learning for Sparse Coding"
   <https://www.di.ens.fr/~fbach/mairal_icml09.pdf>`_
   J. Mairal, F. Bach, J. Ponce, G. Sapiro, 2009
.. [Jen09] `"Structured Sparse Principal Component Analysis"
   <https://www.di.ens.fr/~fbach/sspca_AISTATS2010.pdf>`_
   R. Jenatton, G. Obozinski, F. Bach, 2009

.. _kernel_PCA:

Kernel Principal Component Analysis (kPCA)
==========================================

Exact Kernel PCA
----------------

:class:`KernelPCA` is an extension of PCA which achieves non-linear
dimensionality reduction through the use of kernels (see :ref:`metrics`) [Scholkopf1997]_. It
has many applications including denoising, compression and structured
prediction (kernel dependency estimation). :class:`KernelPCA` supports both
``transform`` and ``inverse_transform``.

.. figure:: ../auto_examples/decomposition/images/sphx_glr_plot_kernel_pca_002.png
    :target: ../auto_examples/decomposition/plot_kernel_pca.html
    :align: center
    :scale: 75%

.. note::
    :meth:`KernelPCA.inverse_transform` relies on a kernel ridge to learn the
    function mapping samples from the PCA basis into the original feature
    space [Bakir2003]_. Thus, the reconstruction obtained with
    :meth:`KernelPCA.inverse_transform` is an approximation. See the example
    linked below for more details.

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_decomposition_plot_kernel_pca.py`
* :ref:`sphx_glr_auto_examples_applications_plot_digits_denoising.py`

.. rubric:: References

.. [Scholkopf1997] Schölkopf, Bernhard, Alexander Smola, and Klaus-Robert Müller.
   `"Kernel principal component analysis."
   <https://people.eecs.berkeley.edu/~wainwrig/stat241b/scholkopf_kernel.pdf>`_
   International conference on artificial neural networks.
   Springer, Berlin, Heidelberg, 1997.

.. [Bakir2003] Bakır, Gökhan H., Jason Weston, and Bernhard Schölkopf.
   `"Learning to find pre-images."
   <https://papers.nips.cc/paper/2003/file/ac1ad983e08ad3304a97e147f522747e-Paper.pdf>`_
   Advances in neural information processing systems 16 (2003): 449-456.

.. _kPCA_Solvers:

Choice of solver for Kernel PCA
-------------------------------

While in :class:`PCA` the number of components is bounded by the number of
features, in :class:`KernelPCA` the number of components is bounded by the
number of samples. Many real-world datasets have large number of samples! In
these cases finding *all* the components with a full kPCA is a waste of
computation time, as data is mostly described by the first few components
(e.g. ``n_components<=100``). 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.

.. dropdown:: Eigensolvers

    The optional parameter ``eigen_solver='randomized'`` can be used to
    *significantly* reduce the computation time when the number of requested
    ``n_components`` is small compared with the number of samples. It relies on
    randomized decomposition methods to find an approximate solution in a shorter
    time.

    The time complexity of the randomized :class:`KernelPCA` is
    :math:`O(n_{\mathrm{samples}}^2 \cdot n_{\mathrm{components}})`
    instead of :math:`O(n_{\mathrm{samples}}^3)` for the exact method
    implemented with ``eigen_solver='dense'``.

    The memory footprint of randomized :class:`KernelPCA` is also proportional to
    :math:`2 \cdot n_{\mathrm{samples}} \cdot n_{\mathrm{components}}` instead of
    :math:`n_{\mathrm{samples}}^2` for the exact method.

    Note: this technique is the same as in :ref:`RandomizedPCA`.

    In addition to the above two solvers, ``eigen_solver='arpack'`` 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 :class:`KernelPCA` for details.

    .. rubric:: References

    * *dense* solver:
      `scipy.linalg.eigh documentation
      <https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.eigh.html>`_

    * *randomized* solver:

      * Algorithm 4.3 in
        :arxiv:`"Finding structure with randomness: Stochastic
        algorithms for constructing approximate matrix decompositions" <0909.4061>`
        Halko, et al. (2009)

      * :arxiv:`"An implementation of a randomized algorithm
        for principal component analysis" <1412.3510>`
        A. Szlam et al. (2014)

    * *arpack* solver:
      `scipy.sparse.linalg.eigsh documentation
      <https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.eigsh.html>`_
      R. B. Lehoucq, D. C. Sorensen, and C. Yang, (1998)

.. _LSA:

Truncated singular value decomposition and latent semantic analysis
===================================================================

:class:`TruncatedSVD` implements a variant of singular value decomposition
(SVD) that only computes the :math:`k` largest singular values,
where :math:`k` is a user-specified parameter.

:class:`TruncatedSVD` is very similar to :class:`PCA`, but differs
in that the matrix :math:`X` does not need to be centered.
When the columnwise (per-feature) means of :math:`X`
are subtracted from the feature values,
truncated SVD on the resulting matrix is equivalent to PCA.

.. dropdown:: About truncated SVD and latent semantic analysis (LSA)

    When truncated SVD is applied to term-document matrices
    (as returned by :class:`~sklearn.feature_extraction.text.CountVectorizer` or
    :class:`~sklearn.feature_extraction.text.TfidfVectorizer`),
    this transformation is known as
    `latent semantic analysis <https://nlp.stanford.edu/IR-book/pdf/18lsi.pdf>`_
    (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.

    .. note::
        LSA is also known as latent semantic indexing, LSI,
        though strictly that refers to its use in persistent indexes
        for information retrieval purposes.

    Mathematically, truncated SVD applied to training samples :math:`X`
    produces a low-rank approximation :math:`X`:

    .. math::
        X \approx X_k = U_k \Sigma_k V_k^\top

    After this operation, :math:`U_k \Sigma_k`
    is the transformed training set with :math:`k` features
    (called ``n_components`` in the API).

    To also transform a test set :math:`X`, we multiply it with :math:`V_k`:

    .. math::
        X' = X V_k

    .. note::
        Most treatments of LSA in the natural language processing (NLP)
        and information retrieval (IR) literature
        swap the axes of the matrix :math:`X` so that it has shape
        ``(n_features, n_samples)``.
        We present LSA in a different way that matches the scikit-learn API better,
        but the singular values found are the same.

    While the :class:`TruncatedSVD` 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 (``sublinear_tf=True, use_idf=True``)
    to bring the feature values closer to a Gaussian distribution,
    compensating for LSA's erroneous assumptions about textual data.

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_text_plot_document_clustering.py`

.. rubric:: References

* Christopher D. Manning, Prabhakar Raghavan and Hinrich Schütze (2008),
  *Introduction to Information Retrieval*, Cambridge University Press,
  chapter 18: `Matrix decompositions & latent semantic indexing
  <https://nlp.stanford.edu/IR-book/pdf/18lsi.pdf>`_

.. _DictionaryLearning:

Dictionary Learning
===================

.. _SparseCoder:

Sparse coding with a precomputed dictionary
-------------------------------------------

The :class:`SparseCoder` 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 ``fit`` 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 ``transform_method`` initialization parameter:

* Orthogonal matching pursuit (:ref:`omp`)

* Least-angle regression (:ref:`least_angle_regression`)

* Lasso computed by least-angle regression

* Lasso using coordinate descent (:ref:`lasso`)

* Thresholding

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.

The dictionary learning objects offer, via the ``split_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.

The split code for a single sample has length ``2 * n_components``
and is constructed using the following rule: First, the regular code of length
``n_components`` is computed. Then, the first ``n_components`` entries of the
``split_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.

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_decomposition_plot_sparse_coding.py`

Generic dictionary learning
---------------------------

Dictionary learning (:class:`DictionaryLearning`) is a matrix factorization
problem that amounts to finding a (usually overcomplete) dictionary that will
perform well at sparsely encoding the fitted data.

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.

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.

.. math::
   (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}}

.. |pca_img2| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_002.png
   :target: ../auto_examples/decomposition/plot_faces_decomposition.html
   :scale: 60%

.. |dict_img2| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_007.png
   :target: ../auto_examples/decomposition/plot_faces_decomposition.html
   :scale: 60%

.. centered:: |pca_img2| |dict_img2|

:math:`||.||_{\text{Fro}}` stands for the Frobenius norm and :math:`||.||_{1,1}`
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 :ref:`SparseCoder`).

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.

.. |dict_img_pos1| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_010.png
    :target: ../auto_examples/decomposition/plot_image_denoising.html
    :scale: 60%

.. |dict_img_pos2| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_011.png
    :target: ../auto_examples/decomposition/plot_image_denoising.html
    :scale: 60%

.. |dict_img_pos3| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_012.png
    :target: ../auto_examples/decomposition/plot_image_denoising.html
    :scale: 60%

.. |dict_img_pos4| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_013.png
    :target: ../auto_examples/decomposition/plot_image_denoising.html
    :scale: 60%

.. centered:: |dict_img_pos1| |dict_img_pos2|
.. centered:: |dict_img_pos3| |dict_img_pos4|

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.

.. figure:: ../auto_examples/decomposition/images/sphx_glr_plot_image_denoising_001.png
    :target: ../auto_examples/decomposition/plot_image_denoising.html
    :align: center
    :scale: 50%

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_decomposition_plot_image_denoising.py`

.. rubric:: References

* `"Online dictionary learning for sparse coding"
  <https://www.di.ens.fr/~fbach/mairal_icml09.pdf>`_
  J. Mairal, F. Bach, J. Ponce, G. Sapiro, 2009

.. _MiniBatchDictionaryLearning:

Mini-batch dictionary learning
------------------------------

:class:`MiniBatchDictionaryLearning` implements a faster, but less accurate
version of the dictionary learning algorithm that is better suited for large
datasets.

By default, :class:`MiniBatchDictionaryLearning` 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.

The estimator also implements ``partial_fit``, 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.

.. currentmodule:: sklearn.cluster

.. image:: ../auto_examples/cluster/images/sphx_glr_plot_dict_face_patches_001.png
    :target: ../auto_examples/cluster/plot_dict_face_patches.html
    :scale: 50%
    :align: right

.. topic:: **Clustering for dictionary learning**

   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 :class:`MiniBatchKMeans` estimator is
   computationally efficient and implements on-line learning with a
   ``partial_fit`` method.

   Example: :ref:`sphx_glr_auto_examples_cluster_plot_dict_face_patches.py`

.. currentmodule:: sklearn.decomposition

.. _FA:

Factor Analysis
===============

In unsupervised learning we only have a dataset :math:`X = \{x_1, x_2, \dots, x_n
\}`. How can this dataset be described mathematically? A very simple
`continuous latent variable` model for :math:`X` is

.. math:: x_i = W h_i + \mu + \epsilon

The vector :math:`h_i` is called "latent" because it is unobserved. :math:`\epsilon` is
considered a noise term distributed according to a Gaussian with mean 0 and
covariance :math:`\Psi` (i.e. :math:`\epsilon \sim \mathcal{N}(0, \Psi)`), :math:`\mu` is some
arbitrary offset vector. Such a model is called "generative" as it describes
how :math:`x_i` is generated from :math:`h_i`. If we use all the :math:`x_i`'s as columns to form
a matrix :math:`\mathbf{X}` and all the :math:`h_i`'s as columns of a matrix :math:`\mathbf{H}`
then we can write (with suitably defined :math:`\mathbf{M}` and :math:`\mathbf{E}`):

.. math::
    \mathbf{X} = W \mathbf{H} + \mathbf{M} + \mathbf{E}

In other words, we *decomposed* matrix :math:`\mathbf{X}`.

If :math:`h_i` is given, the above equation automatically implies the following
probabilistic interpretation:

.. math:: p(x_i|h_i) = \mathcal{N}(Wh_i + \mu, \Psi)

For a complete probabilistic model we also need a prior distribution for the
latent variable :math:`h`. The most straightforward assumption (based on the nice
properties of the Gaussian distribution) is :math:`h \sim \mathcal{N}(0,
\mathbf{I})`.  This yields a Gaussian as the marginal distribution of :math:`x`:

.. math:: p(x) = \mathcal{N}(\mu, WW^T + \Psi)

Now, without any further assumptions the idea of having a latent variable :math:`h`
would be superfluous -- :math:`x` 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 :math:`\Psi`:

* :math:`\Psi = \sigma^2 \mathbf{I}`: This assumption leads to
  the probabilistic model of :class:`PCA`.

* :math:`\Psi = \mathrm{diag}(\psi_1, \psi_2, \dots, \psi_n)`: This model is called
  :class:`FactorAnalysis`, a classical statistical model. The matrix W is
  sometimes called the "factor loading matrix".

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.
:class:`FastICA`) if non-Gaussian priors on the latent variables are assumed.

Factor analysis *can* produce similar components (the columns of its loading
matrix) to :class:`PCA`. However, one can not make any general statements
about these components (e.g. whether they are orthogonal):

.. |pca_img3| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_002.png
    :target: ../auto_examples/decomposition/plot_faces_decomposition.html
    :scale: 60%

.. |fa_img3| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_008.png
    :target: ../auto_examples/decomposition/plot_faces_decomposition.html
    :scale: 60%

.. centered:: |pca_img3| |fa_img3|

The main advantage for Factor Analysis over :class:`PCA` is that
it can model the variance in every direction of the input space independently
(heteroscedastic noise):

.. figure:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_009.png
    :target: ../auto_examples/decomposition/plot_faces_decomposition.html
    :align: center
    :scale: 75%

This allows better model selection than probabilistic PCA in the presence
of heteroscedastic noise:

.. figure:: ../auto_examples/decomposition/images/sphx_glr_plot_pca_vs_fa_model_selection_002.png
    :target: ../auto_examples/decomposition/plot_pca_vs_fa_model_selection.html
    :align: center
    :scale: 75%

Factor Analysis is often followed by a rotation of the factors (with the
parameter `rotation`), 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.

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_decomposition_plot_varimax_fa.py`
* :ref:`sphx_glr_auto_examples_decomposition_plot_pca_vs_fa_model_selection.py`

.. _ICA:

Independent component analysis (ICA)
====================================

Independent component analysis separates a multivariate signal into
additive subcomponents that are maximally independent. It is
implemented in scikit-learn using the :class:`Fast ICA <FastICA>`
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.

It is classically used to separate mixed signals (a problem known as
*blind source separation*), as in the example below:

.. figure:: ../auto_examples/decomposition/images/sphx_glr_plot_ica_blind_source_separation_001.png
    :target: ../auto_examples/decomposition/plot_ica_blind_source_separation.html
    :align: center
    :scale: 60%

ICA can also be used as yet another non linear decomposition that finds
components with some sparsity:

.. |pca_img4| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_002.png
    :target: ../auto_examples/decomposition/plot_faces_decomposition.html
    :scale: 60%

.. |ica_img4| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_004.png
    :target: ../auto_examples/decomposition/plot_faces_decomposition.html
    :scale: 60%

.. centered:: |pca_img4| |ica_img4|

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_decomposition_plot_ica_blind_source_separation.py`
* :ref:`sphx_glr_auto_examples_decomposition_plot_ica_vs_pca.py`
* :ref:`sphx_glr_auto_examples_decomposition_plot_faces_decomposition.py`

.. _NMF:

Non-negative matrix factorization (NMF or NNMF)
===============================================

NMF with the Frobenius norm
---------------------------

:class:`NMF` [1]_ is an alternative approach to decomposition that assumes that the
data and the components are non-negative. :class:`NMF` can be plugged in
instead of :class:`PCA` or its variants, in the cases where the data matrix
does not contain negative values. It finds a decomposition of samples
:math:`X` into two matrices :math:`W` and :math:`H` of non-negative elements,
by optimizing the distance :math:`d` between :math:`X` and the matrix product
:math:`WH`. The most widely used distance function is the squared Frobenius
norm, which is an obvious extension of the Euclidean norm to matrices:

.. 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

Unlike :class:`PCA`, 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.

It has been observed in [Hoyer, 2004] [2]_ that, when carefully constrained,
:class:`NMF` can produce a parts-based representation of the dataset,
resulting in interpretable models. The following example displays 16
sparse components found by :class:`NMF` from the images in the Olivetti
faces dataset, in comparison with the PCA eigenfaces.

.. |pca_img5| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_002.png
    :target: ../auto_examples/decomposition/plot_faces_decomposition.html
    :scale: 60%

.. |nmf_img5| image:: ../auto_examples/decomposition/images/sphx_glr_plot_faces_decomposition_003.png
    :target: ../auto_examples/decomposition/plot_faces_decomposition.html
    :scale: 60%

.. centered:: |pca_img5| |nmf_img5|

The `init` attribute determines the initialization method applied, which
has a great impact on the performance of the method. :class:`NMF` implements the
method Nonnegative Double Singular Value Decomposition. NNDSVD [4]_ 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.

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.

:class:`NMF` can also be initialized with correctly scaled random non-negative
matrices by setting `init="random"`. An integer seed or a
``RandomState`` can also be passed to `random_state` to control
reproducibility.

In :class:`NMF`, 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 :class:`~sklearn.linear_model.ElasticNet`,
we control the combination of L1 and L2 with the `l1_ratio` (:math:`\rho`)
parameter, and the intensity of the regularization with the `alpha_W` and
`alpha_H` (:math:`\alpha_W` and :math:`\alpha_H`) parameters. The priors are
scaled by the number of samples (:math:`n\_samples`) for `H` and the number of
features (:math:`n\_features`) for `W` 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:

.. math::
    (\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

and the regularized objective function is:

.. math::
    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

NMF with a beta-divergence
--------------------------

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:

.. math::
    d_{\mathrm{Fro}}(X, Y) = \frac{1}{2} ||X - Y||_{Fro}^2 = \frac{1}{2} \sum_{i,j} (X_{ij} - {Y}_{ij})^2

Other distance functions can be used in NMF as, for example, the (generalized)
Kullback-Leibler (KL) divergence, also referred as I-divergence:

.. math::
    d_{KL}(X, Y) = \sum_{i,j} (X_{ij} \log(\frac{X_{ij}}{Y_{ij}}) - X_{ij} + Y_{ij})

Or, the Itakura-Saito (IS) divergence:

.. math::
    d_{IS}(X, Y) = \sum_{i,j} (\frac{X_{ij}}{Y_{ij}} - \log(\frac{X_{ij}}{Y_{ij}}) - 1)

These three distances are special cases of the beta-divergence family, with
:math:`\beta = 2, 1, 0` respectively [6]_. The beta-divergence are
defined by :

.. 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})

.. image:: ../images/beta_divergence.png
    :align: center
    :scale: 75%

Note that this definition is not valid if :math:`\beta \in (0; 1)`, yet it can
be continuously extended to the definitions of :math:`d_{KL}` and :math:`d_{IS}`
respectively.

.. dropdown:: NMF implemented solvers

    :class:`NMF` implements two solvers, using Coordinate Descent ('cd') [5]_, and
    Multiplicative Update ('mu') [6]_. The 'mu' solver can optimize every
    beta-divergence, including of course the Frobenius norm (:math:`\beta=2`), the
    (generalized) Kullback-Leibler divergence (:math:`\beta=1`) and the
    Itakura-Saito divergence (:math:`\beta=0`). Note that for
    :math:`\beta \in (1; 2)`, the 'mu' solver is significantly faster than for other
    values of :math:`\beta`. Note also that with a negative (or 0, i.e.
    'itakura-saito') :math:`\beta`, the input matrix cannot contain zero values.

    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.

NMF is best used with the ``fit_transform`` method, which returns the matrix W.
The matrix H is stored into the fitted model in the ``components_`` attribute;
the method ``transform`` will decompose a new matrix X_new based on these
stored components::

    >>> import numpy as np
    >>> X = np.array([[1, 1], [2, 1], [3, 1.2], [4, 1], [5, 0.8], [6, 1]])
    >>> from sklearn.decomposition import NMF
    >>> model = NMF(n_components=2, init='random', random_state=0)
    >>> W = model.fit_transform(X)
    >>> H = model.components_
    >>> X_new = np.array([[1, 0], [1, 6.1], [1, 0], [1, 4], [3.2, 1], [0, 4]])
    >>> W_new = model.transform(X_new)

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_decomposition_plot_faces_decomposition.py`
* :ref:`sphx_glr_auto_examples_applications_plot_topics_extraction_with_nmf_lda.py`

.. _MiniBatchNMF:

Mini-batch Non Negative Matrix Factorization
--------------------------------------------

:class:`MiniBatchNMF` [7]_ implements a faster, but less accurate version of the
non negative matrix factorization (i.e. :class:`~sklearn.decomposition.NMF`),
better suited for large datasets.

By default, :class:`MiniBatchNMF` 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 ``batch_size`` parameter controls
the size of the batches.

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 ``forget_factor``
parameter.

The estimator also implements ``partial_fit``, which updates ``H`` 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.

.. rubric:: References

.. [1] `"Learning the parts of objects by non-negative matrix factorization"
  <http://www.cs.columbia.edu/~blei/fogm/2020F/readings/LeeSeung1999.pdf>`_
  D. Lee, S. Seung, 1999

.. [2] `"Non-negative Matrix Factorization with Sparseness Constraints"
  <https://www.jmlr.org/papers/volume5/hoyer04a/hoyer04a.pdf>`_
  P. Hoyer, 2004

.. [4] `"SVD based initialization: A head start for nonnegative
  matrix factorization"
  <https://www.boutsidis.org/Boutsidis_PRE_08.pdf>`_
  C. Boutsidis, E. Gallopoulos, 2008

.. [5] `"Fast local algorithms for large scale nonnegative matrix and tensor
  factorizations."
  <https://www.researchgate.net/profile/Anh-Huy-Phan/publication/220241471_Fast_Local_Algorithms_for_Large_Scale_Nonnegative_Matrix_and_Tensor_Factorizations>`_
  A. Cichocki, A. Phan, 2009

.. [6] :arxiv:`"Algorithms for nonnegative matrix factorization with
  the beta-divergence" <1010.1763>`
  C. Fevotte, J. Idier, 2011

.. [7] :arxiv:`"Online algorithms for nonnegative matrix factorization with the
  Itakura-Saito divergence" <1106.4198>`
  A. Lefevre, F. Bach, C. Fevotte, 2011

.. _LatentDirichletAllocation:

Latent Dirichlet Allocation (LDA)
=================================

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.

The graphical model of LDA is a three-level generative model:

.. image:: ../images/lda_model_graph.png
   :align: center

Note on notations presented in the graphical model above, which can be found in
Hoffman et al. (2013):

* The corpus is a collection of :math:`D` documents.
* A document is a sequence of :math:`N` words.
* There are :math:`K` topics in the corpus.
* The boxes represent repeated sampling.

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.

.. dropdown:: Details on modeling text corpora

    When modeling text corpora, the model assumes the following generative process
    for a corpus with :math:`D` documents and :math:`K` topics, with :math:`K`
    corresponding to `n_components` in the API:

    1. For each topic :math:`k \in K`, draw :math:`\beta_k \sim
       \mathrm{Dirichlet}(\eta)`. This provides a distribution over the words,
       i.e. the probability of a word appearing in topic :math:`k`.
       :math:`\eta` corresponds to `topic_word_prior`.

    2. For each document :math:`d \in D`, draw the topic proportions
       :math:`\theta_d \sim \mathrm{Dirichlet}(\alpha)`. :math:`\alpha`
       corresponds to `doc_topic_prior`.

    3. For each word :math:`i` in document :math:`d`:

       a. Draw the topic assignment :math:`z_{di} \sim \mathrm{Multinomial}
          (\theta_d)`
       b. Draw the observed word :math:`w_{ij} \sim \mathrm{Multinomial}
          (\beta_{z_{di}})`

    For parameter estimation, the posterior distribution is:

    .. math::
        p(z, \theta, \beta |w, \alpha, \eta) =
        \frac{p(z, \theta, \beta|\alpha, \eta)}{p(w|\alpha, \eta)}

    Since the posterior is intractable, variational Bayesian method
    uses a simpler distribution :math:`q(z,\theta,\beta | \lambda, \phi, \gamma)`
    to approximate it, and those variational parameters :math:`\lambda`,
    :math:`\phi`, :math:`\gamma` are optimized to maximize the Evidence
    Lower Bound (ELBO):

    .. 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)]

    Maximizing ELBO is equivalent to minimizing the Kullback-Leibler(KL) divergence
    between :math:`q(z,\theta,\beta)` and the true posterior
    :math:`p(z, \theta, \beta |w, \alpha, \eta)`.

:class:`LatentDirichletAllocation` 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.

.. note::

  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.

When :class:`LatentDirichletAllocation` 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 `components_` in the model, "document-topic" matrix
can be calculated from ``transform`` method.

:class:`LatentDirichletAllocation` also implements ``partial_fit`` method. This is used
when data can be fetched sequentially.

.. rubric:: Examples

* :ref:`sphx_glr_auto_examples_applications_plot_topics_extraction_with_nmf_lda.py`

.. rubric:: References

* `"Latent Dirichlet Allocation"
  <https://www.jmlr.org/papers/volume3/blei03a/blei03a.pdf>`_
  D. Blei, A. Ng, M. Jordan, 2003

* `"Online Learning for Latent Dirichlet Allocation”
  <https://papers.nips.cc/paper/3902-online-learning-for-latent-dirichlet-allocation.pdf>`_
  M. Hoffman, D. Blei, F. Bach, 2010

* `"Stochastic Variational Inference"
  <https://www.cs.columbia.edu/~blei/papers/HoffmanBleiWangPaisley2013.pdf>`_
  M. Hoffman, D. Blei, C. Wang, J. Paisley, 2013

* `"The varimax criterion for analytic rotation in factor analysis"
  <https://link.springer.com/article/10.1007%2FBF02289233>`_
  H. F. Kaiser, 1958

See also :ref:`nca_dim_reduction` for dimensionality reduction with
Neighborhood Components Analysis.

decompositions                                                                      Decomposing signals in components  matrix factorization problems                                                                        currentmodule   sklearn decomposition       PCA    Principal component analysis  PCA                                      Exact PCA and probabilistic interpretation                                             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   class  PCA  is implemented as a  transformer  object that learns  math  n  components in its   fit   method  and can be used on new data to project it on these components   PCA centers but does not scale the input data for each feature before applying the SVD  The optional parameter   whiten True   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   Below is an example of the iris dataset  which is comprised of 4 features  projected on the 2 dimensions that explain most variance      figure      auto examples decomposition images sphx glr plot pca vs lda 001 png      target     auto examples decomposition plot pca vs lda html      align  center      scale  75    The  class  PCA  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  term  score  method that can be used in cross validation      figure      auto examples decomposition images sphx glr plot pca vs fa model selection 001 png      target     auto examples decomposition plot pca vs fa model selection html      align  center      scale  75       rubric   Examples     ref  sphx glr auto examples decomposition plot pca iris py     ref  sphx glr auto examples decomposition plot pca vs lda py     ref  sphx glr auto examples decomposition plot pca vs fa model selection py        IncrementalPCA   Incremental PCA                  The  class  PCA  object is very useful  but has certain limitations for large datasets  The biggest limitation is that  class  PCA  only supports batch processing  which means all of the data to be processed must fit in main memory  The  class  IncrementalPCA  object uses a different form of processing and allows for partial computations which almost exactly match the results of  class  PCA  while processing the data in a minibatch fashion   class  IncrementalPCA  makes it possible to implement out of core Principal Component Analysis either by     Using its   partial fit   method on chunks of data fetched sequentially   from the local hard drive or a network database     Calling its fit method on a memory mapped file using     numpy memmap      class  IncrementalPCA  only stores estimates of component and noise variances  in order update   explained variance ratio    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   As in  class  PCA    class  IncrementalPCA  centers but does not scale the input data for each feature before applying the SVD      figure      auto examples decomposition images sphx glr plot incremental pca 001 png      target     auto examples decomposition plot incremental pca html      align  center      scale  75      figure      auto examples decomposition images sphx glr plot incremental pca 002 png      target     auto examples decomposition plot incremental pca html      align  center      scale  75       rubric   Examples     ref  sphx glr auto examples decomposition plot incremental pca py        RandomizedPCA   PCA using randomized SVD                           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   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   The class  class  PCA  used with the optional parameter   svd solver  randomized    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   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  math  n  samples    400  and  math  n  features    64  times 64   4096   the computation time is less than 1s       orig img  image      auto examples decomposition images sphx glr plot faces decomposition 001 png     target     auto examples decomposition plot faces decomposition html     scale  60       pca img  image      auto examples decomposition images sphx glr plot faces decomposition 002 png     target     auto examples decomposition plot faces decomposition html     scale  60      centered    orig img   pca img   If we note  math  n   max     max n   mathrm samples    n   mathrm features     and  math  n   min     min n   mathrm samples    n   mathrm features      the time complexity of the randomized  class  PCA  is  math  O n   max  2  cdot n   mathrm components     instead of  math  O n   max  2  cdot n   min    for the exact method implemented in  class  PCA    The memory footprint of randomized  class  PCA  is also proportional to  math  2  cdot n   max   cdot n   mathrm components    instead of  math  n   max   cdot n   min   for the exact method   Note  the implementation of   inverse transform   in  class  PCA  with   svd solver  randomized    is not the exact inverse transform of   transform   even when   whiten False    default        rubric   Examples     ref  sphx glr auto examples applications plot face recognition py     ref  sphx glr auto examples decomposition plot faces decomposition py      rubric   References    Algorithm 4 3 in    arxiv   Finding structure with randomness  Stochastic algorithms for   constructing approximate matrix decompositions   0909 4061     Halko  et al   2009     arxiv   An implementation of a randomized algorithm for principal component   analysis   1412 3510   A  Szlam et al  2014      SparsePCA   Sparse principal components analysis  SparsePCA and MiniBatchSparsePCA                                                                            class  SparsePCA  is a variant of PCA  with the goal of extracting the set of sparse components that best reconstruct the data   Mini batch sparse PCA   class  MiniBatchSparsePCA   is a variant of  class  SparsePCA  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    Principal component analysis   class  PCA   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   Sparse principal components yields a more parsimonious  interpretable representation  clearly emphasizing which of the original features contribute to the differences between samples   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  math  h  in  mathbf R   4096    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  Jen09   for a review of such methods  For more details on how to use Sparse PCA  see the Examples section  below        spca img  image      auto examples decomposition images sphx glr plot faces decomposition 005 png     target     auto examples decomposition plot faces decomposition html     scale  60      centered    pca img   spca img   Note that there are many different formulations for the Sparse PCA problem  The one implemented here is based on  Mrl09     The optimization problem solved is a PCA problem  dictionary learning  with an  math   ell 1  penalty on the components      math       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    math          text Fro    stands for the Frobenius norm and  math         1 1   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  math         1 1   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   alpha    Small values lead to a gently regularized factorization  while larger values shrink many coefficients to zero      note      While in the spirit of an online algorithm  the class    class  MiniBatchSparsePCA  does not implement   partial fit   because   the algorithm is online along the features direction  not the samples   direction      rubric   Examples     ref  sphx glr auto examples decomposition plot faces decomposition py      rubric   References      Mrl09    Online Dictionary Learning for Sparse Coding      https   www di ens fr  fbach mairal icml09 pdf       J  Mairal  F  Bach  J  Ponce  G  Sapiro  2009     Jen09    Structured Sparse Principal Component Analysis      https   www di ens fr  fbach sspca AISTATS2010 pdf       R  Jenatton  G  Obozinski  F  Bach  2009       kernel PCA   Kernel Principal Component Analysis  kPCA                                              Exact Kernel PCA                    class  KernelPCA  is an extension of PCA which achieves non linear dimensionality reduction through the use of kernels  see  ref  metrics    Scholkopf1997    It has many applications including denoising  compression and structured prediction  kernel dependency estimation    class  KernelPCA  supports both   transform   and   inverse transform        figure      auto examples decomposition images sphx glr plot kernel pca 002 png      target     auto examples decomposition plot kernel pca html      align  center      scale  75      note        meth  KernelPCA inverse transform  relies on a kernel ridge to learn the     function mapping samples from the PCA basis into the original feature     space  Bakir2003    Thus  the reconstruction obtained with      meth  KernelPCA inverse transform  is an approximation  See the example     linked below for more details      rubric   Examples     ref  sphx glr auto examples decomposition plot kernel pca py     ref  sphx glr auto examples applications plot digits denoising py      rubric   References      Scholkopf1997  Sch lkopf  Bernhard  Alexander Smola  and Klaus Robert M ller       Kernel principal component analysis       https   people eecs berkeley edu  wainwrig stat241b scholkopf kernel pdf       International conference on artificial neural networks     Springer  Berlin  Heidelberg  1997       Bakir2003  Bak r  G khan H   Jason Weston  and Bernhard Sch lkopf       Learning to find pre images       https   papers nips cc paper 2003 file ac1ad983e08ad3304a97e147f522747e Paper pdf       Advances in neural information processing systems 16  2003   449 456       kPCA Solvers   Choice of solver for Kernel PCA                                  While in  class  PCA  the number of components is bounded by the number of features  in  class  KernelPCA  the number of components is bounded by the number of samples  Many real world datasets have large number of samples  In these cases finding  all  the components with a full kPCA is a waste of computation time  as data is mostly described by the first few components  e g    n components  100     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       dropdown   Eigensolvers      The optional parameter   eigen solver  randomized    can be used to      significantly  reduce the computation time when the number of requested       n components   is small compared with the number of samples  It relies on     randomized decomposition methods to find an approximate solution in a shorter     time       The time complexity of the randomized  class  KernelPCA  is      math  O n   mathrm samples   2  cdot n   mathrm components         instead of  math  O n   mathrm samples   3   for the exact method     implemented with   eigen solver  dense          The memory footprint of randomized  class  KernelPCA  is also proportional to      math  2  cdot n   mathrm samples    cdot n   mathrm components    instead of      math  n   mathrm samples   2  for the exact method       Note  this technique is the same as in  ref  RandomizedPCA        In addition to the above two solvers    eigen solver  arpack    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  class  KernelPCA  for details          rubric   References         dense  solver         scipy linalg eigh documentation        https   docs scipy org doc scipy reference generated scipy linalg eigh html            randomized  solver           Algorithm 4 3 in          arxiv   Finding structure with randomness  Stochastic         algorithms for constructing approximate matrix decompositions   0909 4061           Halko  et al   2009            arxiv   An implementation of a randomized algorithm         for principal component analysis   1412 3510           A  Szlam et al   2014          arpack  solver         scipy sparse linalg eigsh documentation        https   docs scipy org doc scipy reference generated scipy sparse linalg eigsh html          R  B  Lehoucq  D  C  Sorensen  and C  Yang   1998        LSA   Truncated singular value decomposition and latent semantic analysis                                                                       class  TruncatedSVD  implements a variant of singular value decomposition  SVD  that only computes the  math  k  largest singular values  where  math  k  is a user specified parameter    class  TruncatedSVD  is very similar to  class  PCA   but differs in that the matrix  math  X  does not need to be centered  When the columnwise  per feature  means of  math  X  are subtracted from the feature values  truncated SVD on the resulting matrix is equivalent to PCA      dropdown   About truncated SVD and latent semantic analysis  LSA       When truncated SVD is applied to term document matrices      as returned by  class   sklearn feature extraction text CountVectorizer  or      class   sklearn feature extraction text TfidfVectorizer        this transformation is known as      latent semantic analysis  https   nlp stanford edu IR book pdf 18lsi pdf         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          note           LSA is also known as latent semantic indexing  LSI          though strictly that refers to its use in persistent indexes         for information retrieval purposes       Mathematically  truncated SVD applied to training samples  math  X      produces a low rank approximation  math  X           math           X  approx X k   U k  Sigma k V k  top      After this operation   math  U k  Sigma k      is the transformed training set with  math  k  features      called   n components   in the API        To also transform a test set  math  X   we multiply it with  math  V k           math           X    X V k         note           Most treatments of LSA in the natural language processing  NLP          and information retrieval  IR  literature         swap the axes of the matrix  math  X  so that it has shape            n features  n samples             We present LSA in a different way that matches the scikit learn API better          but the singular values found are the same       While the  class  TruncatedSVD  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    sublinear tf True  use idf True        to bring the feature values closer to a Gaussian distribution      compensating for LSA s erroneous assumptions about textual data      rubric   Examples     ref  sphx glr auto examples text plot document clustering py      rubric   References    Christopher D  Manning  Prabhakar Raghavan and Hinrich Sch tze  2008      Introduction to Information Retrieval   Cambridge University Press    chapter 18   Matrix decompositions   latent semantic indexing    https   nlp stanford edu IR book pdf 18lsi pdf           DictionaryLearning   Dictionary Learning                          SparseCoder   Sparse coding with a precomputed dictionary                                              The  class  SparseCoder  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   fit   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   transform method   initialization parameter     Orthogonal matching pursuit   ref  omp      Least angle regression   ref  least angle regression      Lasso computed by least angle regression    Lasso using coordinate descent   ref  lasso      Thresholding  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   The dictionary learning objects offer  via the   split 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   The split code for a single sample has length   2   n components   and is constructed using the following rule  First  the regular code of length   n components   is computed  Then  the first   n components   entries of the   split 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       rubric   Examples     ref  sphx glr auto examples decomposition plot sparse coding py    Generic dictionary learning                              Dictionary learning   class  DictionaryLearning   is a matrix factorization problem that amounts to finding a  usually overcomplete  dictionary that will perform well at sparsely encoding the fitted data   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   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      math       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         pca img2  image      auto examples decomposition images sphx glr plot faces decomposition 002 png     target     auto examples decomposition plot faces decomposition html     scale  60       dict img2  image      auto examples decomposition images sphx glr plot faces decomposition 007 png     target     auto examples decomposition plot faces decomposition html     scale  60      centered    pca img2   dict img2    math          text Fro    stands for the Frobenius norm and  math         1 1   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  ref  SparseCoder     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        dict img pos1  image      auto examples decomposition images sphx glr plot faces decomposition 010 png      target     auto examples decomposition plot image denoising html      scale  60       dict img pos2  image      auto examples decomposition images sphx glr plot faces decomposition 011 png      target     auto examples decomposition plot image denoising html      scale  60       dict img pos3  image      auto examples decomposition images sphx glr plot faces decomposition 012 png      target     auto examples decomposition plot image denoising html      scale  60       dict img pos4  image      auto examples decomposition images sphx glr plot faces decomposition 013 png      target     auto examples decomposition plot image denoising html      scale  60      centered    dict img pos1   dict img pos2     centered    dict img pos3   dict img pos4    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       figure      auto examples decomposition images sphx glr plot image denoising 001 png      target     auto examples decomposition plot image denoising html      align  center      scale  50       rubric   Examples     ref  sphx glr auto examples decomposition plot image denoising py       rubric   References      Online dictionary learning for sparse coding     https   www di ens fr  fbach mairal icml09 pdf      J  Mairal  F  Bach  J  Ponce  G  Sapiro  2009      MiniBatchDictionaryLearning   Mini batch dictionary learning                                  class  MiniBatchDictionaryLearning  implements a faster  but less accurate version of the dictionary learning algorithm that is better suited for large datasets   By default   class  MiniBatchDictionaryLearning  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   The estimator also implements   partial fit    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      currentmodule   sklearn cluster     image      auto examples cluster images sphx glr plot dict face patches 001 png      target     auto examples cluster plot dict face patches html      scale  50       align  right     topic     Clustering for dictionary learning       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  class  MiniBatchKMeans  estimator is    computationally efficient and implements on line learning with a      partial fit   method      Example   ref  sphx glr auto examples cluster plot dict face patches py      currentmodule   sklearn decomposition      FA   Factor Analysis                  In unsupervised learning we only have a dataset  math  X     x 1  x 2   dots  x n      How can this dataset be described mathematically  A very simple  continuous latent variable  model for  math  X  is     math   x i   W h i    mu    epsilon  The vector  math  h i  is called  latent  because it is unobserved   math   epsilon  is considered a noise term distributed according to a Gaussian with mean 0 and covariance  math   Psi   i e   math   epsilon  sim  mathcal N  0   Psi      math   mu  is some arbitrary offset vector  Such a model is called  generative  as it describes how  math  x i  is generated from  math  h i   If we use all the  math  x i  s as columns to form a matrix  math   mathbf X   and all the  math  h i  s as columns of a matrix  math   mathbf H   then we can write  with suitably defined  math   mathbf M   and  math   mathbf E         math        mathbf X    W  mathbf H     mathbf M     mathbf E   In other words  we  decomposed  matrix  math   mathbf X     If  math  h i  is given  the above equation automatically implies the following probabilistic interpretation      math   p x i h i     mathcal N  Wh i    mu   Psi   For a complete probabilistic model we also need a prior distribution for the latent variable  math  h   The most straightforward assumption  based on the nice properties of the Gaussian distribution  is  math  h  sim  mathcal N  0   mathbf I      This yields a Gaussian as the marginal distribution of  math  x       math   p x     mathcal N   mu  WW T    Psi   Now  without any further assumptions the idea of having a latent variable  math  h  would be superfluous     math  x  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  math   Psi       math   Psi    sigma 2  mathbf I    This assumption leads to   the probabilistic model of  class  PCA       math   Psi    mathrm diag   psi 1   psi 2   dots   psi n    This model is called    class  FactorAnalysis   a classical statistical model  The matrix W is   sometimes called the  factor loading matrix    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   class  FastICA   if non Gaussian priors on the latent variables are assumed   Factor analysis  can  produce similar components  the columns of its loading matrix  to  class  PCA   However  one can not make any general statements about these components  e g  whether they are orthogonal        pca img3  image      auto examples decomposition images sphx glr plot faces decomposition 002 png      target     auto examples decomposition plot faces decomposition html      scale  60       fa img3  image      auto examples decomposition images sphx glr plot faces decomposition 008 png      target     auto examples decomposition plot faces decomposition html      scale  60      centered    pca img3   fa img3   The main advantage for Factor Analysis over  class  PCA  is that it can model the variance in every direction of the input space independently  heteroscedastic noise       figure      auto examples decomposition images sphx glr plot faces decomposition 009 png      target     auto examples decomposition plot faces decomposition html      align  center      scale  75   This allows better model selection than probabilistic PCA in the presence of heteroscedastic noise      figure      auto examples decomposition images sphx glr plot pca vs fa model selection 002 png      target     auto examples decomposition plot pca vs fa model selection html      align  center      scale  75   Factor Analysis is often followed by a rotation of the factors  with the parameter  rotation    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      rubric   Examples     ref  sphx glr auto examples decomposition plot varimax fa py     ref  sphx glr auto examples decomposition plot pca vs fa model selection py        ICA   Independent component analysis  ICA                                        Independent component analysis separates a multivariate signal into additive subcomponents that are maximally independent  It is implemented in scikit learn using the  class  Fast ICA  FastICA   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   It is classically used to separate mixed signals  a problem known as  blind source separation    as in the example below      figure      auto examples decomposition images sphx glr plot ica blind source separation 001 png      target     auto examples decomposition plot ica blind source separation html      align  center      scale  60    ICA can also be used as yet another non linear decomposition that finds components with some sparsity       pca img4  image      auto examples decomposition images sphx glr plot faces decomposition 002 png      target     auto examples decomposition plot faces decomposition html      scale  60       ica img4  image      auto examples decomposition images sphx glr plot faces decomposition 004 png      target     auto examples decomposition plot faces decomposition html      scale  60      centered    pca img4   ica img4      rubric   Examples     ref  sphx glr auto examples decomposition plot ica blind source separation py     ref  sphx glr auto examples decomposition plot ica vs pca py     ref  sphx glr auto examples decomposition plot faces decomposition py        NMF   Non negative matrix factorization  NMF or NNMF                                                   NMF with the Frobenius norm                               class  NMF   1   is an alternative approach to decomposition that assumes that the data and the components are non negative   class  NMF  can be plugged in instead of  class  PCA  or its variants  in the cases where the data matrix does not contain negative values  It finds a decomposition of samples  math  X  into two matrices  math  W  and  math  H  of non negative elements  by optimizing the distance  math  d  between  math  X  and the matrix product  math  WH   The most widely used distance function is the squared Frobenius norm  which is an obvious extension of the Euclidean norm to matrices      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  Unlike  class  PCA   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   It has been observed in  Hoyer  2004   2   that  when carefully constrained   class  NMF  can produce a parts based representation of the dataset  resulting in interpretable models  The following example displays 16 sparse components found by  class  NMF  from the images in the Olivetti faces dataset  in comparison with the PCA eigenfaces       pca img5  image      auto examples decomposition images sphx glr plot faces decomposition 002 png      target     auto examples decomposition plot faces decomposition html      scale  60       nmf img5  image      auto examples decomposition images sphx glr plot faces decomposition 003 png      target     auto examples decomposition plot faces decomposition html      scale  60      centered    pca img5   nmf img5    The  init  attribute determines the initialization method applied  which has a great impact on the performance of the method   class  NMF  implements the method Nonnegative Double Singular Value Decomposition  NNDSVD  4   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   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    class  NMF  can also be initialized with correctly scaled random non negative matrices by setting  init  random    An integer seed or a   RandomState   can also be passed to  random state  to control reproducibility   In  class  NMF   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  class   sklearn linear model ElasticNet   we control the combination of L1 and L2 with the  l1 ratio    math   rho   parameter  and the intensity of the regularization with the  alpha W  and  alpha H    math   alpha W  and  math   alpha H   parameters  The priors are scaled by the number of samples   math  n  samples   for  H  and the number of features   math  n  features   for  W  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      math         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  and the regularized objective function is      math       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  NMF with a beta divergence                             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      math       d   mathrm Fro   X  Y     frac 1  2    X   Y    Fro  2    frac 1  2   sum  i j   X  ij     Y   ij   2  Other distance functions can be used in NMF as  for example  the  generalized  Kullback Leibler  KL  divergence  also referred as I divergence      math       d  KL  X  Y     sum  i j   X  ij   log  frac X  ij   Y  ij      X  ij    Y  ij    Or  the Itakura Saito  IS  divergence      math       d  IS  X  Y     sum  i j    frac X  ij   Y  ij      log  frac X  ij   Y  ij      1   These three distances are special cases of the beta divergence family  with  math   beta   2  1  0  respectively  6    The beta divergence are defined by       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       image      images beta divergence png      align  center      scale  75   Note that this definition is not valid if  math   beta  in  0  1    yet it can be continuously extended to the definitions of  math  d  KL   and  math  d  IS   respectively      dropdown   NMF implemented solvers       class  NMF  implements two solvers  using Coordinate Descent   cd    5    and     Multiplicative Update   mu    6    The  mu  solver can optimize every     beta divergence  including of course the Frobenius norm   math   beta 2    the      generalized  Kullback Leibler divergence   math   beta 1   and the     Itakura Saito divergence   math   beta 0    Note that for      math   beta  in  1  2    the  mu  solver is significantly faster than for other     values of  math   beta   Note also that with a negative  or 0  i e       itakura saito    math   beta   the input matrix cannot contain zero values       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   NMF is best used with the   fit transform   method  which returns the matrix W  The matrix H is stored into the fitted model in the   components    attribute  the method   transform   will decompose a new matrix X new based on these stored components            import numpy as np         X   np array   1  1    2  1    3  1 2    4  1    5  0 8    6  1            from sklearn decomposition import NMF         model   NMF n components 2  init  random   random state 0          W   model fit transform X          H   model components          X new   np array   1  0    1  6 1    1  0    1  4    3 2  1    0  4            W new   model transform X new        rubric   Examples     ref  sphx glr auto examples decomposition plot faces decomposition py     ref  sphx glr auto examples applications plot topics extraction with nmf lda py       MiniBatchNMF   Mini batch Non Negative Matrix Factorization                                                class  MiniBatchNMF   7   implements a faster  but less accurate version of the non negative matrix factorization  i e   class   sklearn decomposition NMF    better suited for large datasets   By default   class  MiniBatchNMF  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   batch size   parameter controls the size of the batches   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   forget factor   parameter   The estimator also implements   partial fit    which updates   H   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      rubric   References      1    Learning the parts of objects by non negative matrix factorization     http   www cs columbia edu  blei fogm 2020F readings LeeSeung1999 pdf      D  Lee  S  Seung  1999      2    Non negative Matrix Factorization with Sparseness Constraints     https   www jmlr org papers volume5 hoyer04a hoyer04a pdf      P  Hoyer  2004      4    SVD based initialization  A head start for nonnegative   matrix factorization     https   www boutsidis org Boutsidis PRE 08 pdf      C  Boutsidis  E  Gallopoulos  2008      5    Fast local algorithms for large scale nonnegative matrix and tensor   factorizations      https   www researchgate net profile Anh Huy Phan publication 220241471 Fast Local Algorithms for Large Scale Nonnegative Matrix and Tensor Factorizations      A  Cichocki  A  Phan  2009      6   arxiv   Algorithms for nonnegative matrix factorization with   the beta divergence   1010 1763     C  Fevotte  J  Idier  2011      7   arxiv   Online algorithms for nonnegative matrix factorization with the   Itakura Saito divergence   1106 4198     A  Lefevre  F  Bach  C  Fevotte  2011      LatentDirichletAllocation   Latent Dirichlet Allocation  LDA                                     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   The graphical model of LDA is a three level generative model      image      images lda model graph png     align  center  Note on notations presented in the graphical model above  which can be found in Hoffman et al   2013      The corpus is a collection of  math  D  documents    A document is a sequence of  math  N  words    There are  math  K  topics in the corpus    The boxes represent repeated sampling   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      dropdown   Details on modeling text corpora      When modeling text corpora  the model assumes the following generative process     for a corpus with  math  D  documents and  math  K  topics  with  math  K      corresponding to  n components  in the API       1  For each topic  math  k  in K   draw  math   beta k  sim         mathrm Dirichlet   eta    This provides a distribution over the words         i e  the probability of a word appearing in topic  math  k           math   eta  corresponds to  topic word prior        2  For each document  math  d  in D   draw the topic proportions         math   theta d  sim  mathrm Dirichlet   alpha     math   alpha         corresponds to  doc topic prior        3  For each word  math  i  in document  math  d           a  Draw the topic assignment  math  z  di   sim  mathrm Multinomial              theta d          b  Draw the observed word  math  w  ij   sim  mathrm Multinomial              beta  z  di          For parameter estimation  the posterior distribution is          math           p z   theta   beta  w   alpha   eta             frac p z   theta   beta  alpha   eta   p w  alpha   eta        Since the posterior is intractable  variational Bayesian method     uses a simpler distribution  math  q z  theta  beta    lambda   phi   gamma       to approximate it  and those variational parameters  math   lambda        math   phi    math   gamma  are optimized to maximize the Evidence     Lower Bound  ELBO           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        Maximizing ELBO is equivalent to minimizing the Kullback Leibler KL  divergence     between  math  q z  theta  beta   and the true posterior      math  p z   theta   beta  w   alpha   eta       class  LatentDirichletAllocation  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      note      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   When  class  LatentDirichletAllocation  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  components   in the model   document topic  matrix can be calculated from   transform   method    class  LatentDirichletAllocation  also implements   partial fit   method  This is used when data can be fetched sequentially      rubric   Examples     ref  sphx glr auto examples applications plot topics extraction with nmf lda py      rubric   References      Latent Dirichlet Allocation     https   www jmlr org papers volume3 blei03a blei03a pdf      D  Blei  A  Ng  M  Jordan  2003      Online Learning for Latent Dirichlet Allocation     https   papers nips cc paper 3902 online learning for latent dirichlet allocation pdf      M  Hoffman  D  Blei  F  Bach  2010      Stochastic Variational Inference     https   www cs columbia edu  blei papers HoffmanBleiWangPaisley2013 pdf      M  Hoffman  D  Blei  C  Wang  J  Paisley  2013      The varimax criterion for analytic rotation in factor analysis     https   link springer com article 10 1007 2FBF02289233      H  F  Kaiser  1958  See also  ref  nca dim reduction  for dimensionality reduction with Neighborhood Components Analysis