Decomposing signals in components (matrix factorization problems)
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 scikitlearn, PCA
is implemented as a transformer object that learns components in its fit
method, and can be used on new data to project it on these components.
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 downstream 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 KMeans 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:
The 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 score method that can be used in crossvalidation:
Incremental PCA
The PCA
object is very useful, but has certain limitations for large datasets. The biggest limitation is that PCA
only supports batch processing, which means all of the data to be processed must fit in main memory. The IncrementalPCA
object uses a different form of processing and allows for partial computations which almost exactly match the results of PCA
while processing the data in a minibatch fashion. IncrementalPCA
makes it possible to implement outofcore 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.
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.
PCA using randomized SVD
It is often interesting to project data to a lowerdimensional 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 graylevel 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 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 and , the computation time is less than 1s:
Note: with the optional parameter
svd_solver='randomized'
, we also need to give PCA the size of the lowerdimensional spaceas a mandatory input parameter.
n_components
If we note and , the time complexity of the randomized PCA is instead of for the exact method implemented in PCA.
The memory footprint of randomized PCA is also proportional to instead of for the exact method.
Note: the implementation of inverse_transform
in PCA with svd_solver='randomized'
is not the exact inverse transform of transform even when whiten=False
(default).
Kernel PCA
KernelPCA
is an extension of PCA which achieves nonlinear dimensionality reduction through the use of kernels (see Pairwise metrics, Affinities and Kernels). It has many applications including denoising, compression and structured prediction (kernel dependency estimation). KernelPCA
supports both transform
and inverse_transform
.
Sparse principal components analysis (SparsePCA and MiniBatchSparsePCA)
SparsePCA
is a variant of PCA, with the goal of extracting the set of sparse components that best reconstruct the data.
Minibatch sparse PCA (MiniBatchSparsePCA
) is a variant of 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 (PCA
) has the disadvantage that the components extracted by this method have exclusively dense expressions, i.e. they have nonzero 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 nonzero coefficients to be vertically adjacent. The model does not enforce this mathematically: each component is a vector , and there is no notion of vertical adjacency except during the humanfriendly 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 sparsityinducing norms that take into account adjacency and different kinds of structure. For more details on how to use Sparse PCA, see the Examples section, below.
Note that there are many different formulations for the Sparse PCA problem. The optimization problem solved is a PCA problem (dictionary learning) with an penalty on the components:
subject to
for all
The sparsityinducing 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
MiniBatchSparsePCA
does not implementpartial_fit
because the algorithm is online along the features direction, not the samples direction.
Truncated singular value decomposition and latent semantic analysis
TruncatedSVD
implements a variant of singular value decomposition (SVD) that only computes the largest singular values, where is a userspecified parameter.
When truncated SVD is applied to termdocument matrices (as returned by CountVectorizer
or TfidfVectorizer
), this transformation is known as latent semantic analysis (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 termdocument 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 produces a lowrank approximation :
After this operation, is the transformed training set with features (called n_components
in the API).
To also transform a test set , we multiply it with :
Note: Most treatments of LSA in the natural language processing (NLP) and information retrieval (IR) literature swap the axes of the matrix so that it has shape
n_features × n_samples
. We present LSA in a different way that matches the scikitlearn API better, but the singular values found are the same.
TruncatedSVD
is very similar to PCA
, but differs in that it works on sample matrices directly instead of their covariance matrices. When the columnwise (perfeature) means of are subtracted from the feature values, truncated SVD on the resulting matrix is equivalent to PCA. In practical terms, this means that the TruncatedSVD
transformer accepts scipy.sparse
matrices without the need to densify them, as densifying may fill up memory even for mediumsized document collections.
While the TruncatedSVD
transformer works with any (sparse) feature matrix, using it on tf–idf matrices is recommended over raw frequency counts in an LSA/document processing setting. In particular, sublinear scaling and inverse document frequency should be turned on (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.
Dictionary Learning
Sparse coding with a precomputed dictionary
The 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 (Orthogonal Matching Pursuit (OMP))
 Leastangle regression (Least Angle Regression)
 Lasso computed by leastangle regression
 Lasso using coordinate descent (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 nonnegative.
Generic dictionary learning
Dictionary learning (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.
subject to
for all
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 Sparse coding with a precomputed dictionary).
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.
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.
Minibatch dictionary learning
MiniBatchDictionaryLearning
implements a faster, but less accurate version of the dictionary learning algorithm that is better suited for large datasets.
By default, MiniBatchDictionaryLearning
divides the data into minibatches and optimizes in an online manner by cycling over the minibatches 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 minibatch. 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.
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
MiniBatchKMeans
estimator is computationally efficient and implements online learning with apartial_fit
method.
Factor Analysis
In unsupervised learning we only have a dataset . How can this dataset be described mathematically? A very simple continuous latent variable model for is
The vector is called “latent” because it is unobserved. is considered a noise term distributed according to a Gaussian with mean 0 and covariance (i.e. ), is some arbitrary offset vector. Such a model is called “generative” as it describes how is generated from . If we use all the ‘s as columns to form a matrix and all the ‘s as columns of a matrix then we can write (with suitably defined and ):
In other words, we decomposed matrix .
If is given, the above equation automatically implies the following probabilistic interpretation:
For a complete probabilistic model we also need a prior distribution for the latent variable . The most straightforward assumption (based on the nice properties of the Gaussian distribution) is . This yields a Gaussian as the marginal distribution of :
Now, without any further assumptions the idea of having a latent variable would be superfluous – 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 :

: This assumption leads to the probabilistic model of
PCA
. 
: This model is called
FactorAnalysis
, a classical statistical model. The matrix W is sometimes called the “factor loading matrix”.
Both models essentially estimate a Gaussian with a lowrank 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. FastICA
) if nonGaussian priors on the latent variables are assumed.
Factor analysis can produce similar components (the columns of its loading matrix) to PCA
. However, one can not make any general statements about these components (e.g. whether they are orthogonal):
The main advantage for Factor Analysis over PCA
is that it can model the variance in every direction of the input space independently (heteroscedastic noise):
This allows better model selection than probabilistic PCA in the presence of heteroscedastic noise:
Independent component analysis (ICA)
Independent component analysis separates a multivariate signal into additive subcomponents that are maximally independent. It is implemented in scikitlearn using the Fast ICA
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:
ICA can also be used as yet another non linear decomposition that finds components with some sparsity:
Nonnegative matrix factorization (NMF or NNMF)
NMF with the Frobenius norm
NMF
is an alternative approach to decomposition that assumes that the data and the components are nonnegative. NMF
can be plugged in instead of PCA
or its variants, in the cases where the data matrix does not contain negative values. It finds a decomposition of samples into two matrices and of nonnegative elements, by optimizing the distance between and the matrix product . The most widely used distance function is the squared Frobenius norm, which is an obvious extension of the Euclidean norm to matrices:
Unlike 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] that, when carefully constrained, NMF
can produce a partsbased representation of the dataset, resulting in interpretable models. The following example displays 16 sparse components found by NMF
from the images in the Olivetti faces dataset, in comparison with the PCA eigenfaces.
The init
attribute determines the initialization method applied, which has a great impact on the performance of the method. NMF
implements the method Nonnegative Double Singular Value Decomposition. NNDSVD 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.
NMF
can also be initialized with correctly scaled random nonnegative matrices by setting init="random"
. An integer seed or a RandomState
can also be passed to random_state
to control reproducibility.
In 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 ElasticNet
, we control the combination of L1 and L2 with the l1_ratio
() parameter, and the intensity of the regularization with the alpha
() parameter. Then the priors terms are:
and the regularized objective function is:
NMF
regularizes both W and H. The public function non_negative_factorization
allows a finer control through the regularization
attribute, and may regularize only W, only H, or both.
NMF with a betadivergence
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:
Other distance functions can be used in NMF as, for example, the (generalized) KullbackLeibler (KL) divergence, also referred as Idivergence:
Or, the ItakuraSaito (IS) divergence:
These three distances are special cases of the betadivergence family, with respectively . The betadivergence are defined by :
Note that this definition is not valid if , yet it can be continuously extended to the definitions of and respectively.
NMF
implements two solvers, using Coordinate Descent (‘cd’), and Multiplicative Update (‘mu’). The ‘mu’ solver can optimize every betadivergence, including of course the Frobenius norm (), the (generalized) KullbackLeibler divergence () and the ItakuraSaito divergence (). Note that for , the ‘mu’ solver is significantly faster than for other values of . Note also that with a negative (or 0, i.e. ‘itakurasaito’) , the input matrix cannot contain zero values.
The ‘cd’ solver can only optimize the Frobenius norm. Due to the underlying nonconvexity 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)
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 threelevel generative model:
Note on notations presented in the graphical model above, which can be found in Hoffman et al. (2013):
 The corpus is a collection of documents .
 A document is a sequence of words.
 There are 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.
When modeling text corpora, the model assumes the following generative process for a corpus with documents and topics, with corresponding to n_components
in the API:

For each topic , draw . This provides a distribution over the words, i.e. the probability of a word appearing in topic . corresponds to
topic_word_prior
. 
For each document , draw the topic proportions . corresponds to
doc_topic_prior
.  For each word in document :
 Draw the topic assignment
 Draw the observed word
For parameter estimation, the posterior distribution is:
Since the posterior is intractable, variational Bayesian method uses a simpler distribution to approximate it, and those variational parameters , , are optimized to maximize the Evidence Lower Bound (ELBO):
Maximizing ELBO is equivalent to minimizing the KullbackLeibler(KL) divergence between and the true posterior .
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 minibatch 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 minibatch size and attributes related to learning rate setting.
When LatentDirichletAllocation
is applied on a “documentterm” matrix, the matrix will be decomposed into a “topicterm” matrix and a “documenttopic” matrix. While “topicterm” matrix is stored as components_
in the model, “documenttopic” matrix can be calculated from transform method.
LatentDirichletAllocation
also implements partial_fit
method. This is used when data can be fetched sequentially.