2.5. Decomposing signals in components (matrix factorization problems)

2.5. Decomposing signals in components (matrix factorization problems)

2.5.1. Principal component analysis (PCA)

2.5.1.1. Exact PCA and probabilistic interpretation

PCA is used to decompose a multivariate dataset in a set of successiveorthogonal components that explain a maximum amount of the variance. Inscikit-learn, PCA is implemented as a transformer objectthat learns

components in its fit method, and can be used on newdata to project it on these components.

PCA centers but does not scale the input data for each feature beforeapplying the SVD. The optional parameter whiten=True makes itpossible to project the data onto the singular space while scaling eachcomponent to unit variance. This is often useful if the models down-stream makestrong assumptions on the isotropy of the signal: this is for example the casefor Support Vector Machines with the RBF kernel and the K-Means clusteringalgorithm.

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

The PCA object also provides aprobabilistic interpretation of the PCA that can give a likelihood ofdata based on the amount of variance it explains. As such it implements ascore method that can be used in cross-validation:

Examples:

2.5.1.2. Incremental PCA

The PCA object is very useful, but has certain limitations forlarge datasets. The biggest limitation is that PCA only supportsbatch processing, which means all of the data to be processed must fit in mainmemory. The IncrementalPCA object uses a different form ofprocessing and allows for partial computations which almostexactly match the results of PCA while processing the data in aminibatch fashion. IncrementalPCA makes it possible to implementout-of-core Principal Component Analysis either by:

Using its partial_fit method on chunks of data fetched sequentiallyfrom the local hard drive or a network database.
Calling its fit method on a sparse matrix or a memory mapped file usingnumpy.memmap.

IncrementalPCA only stores estimates of component and noise variances,in order update explainedvariance_ratio incrementally. This is whymemory usage depends on the number of samples per batch, rather than thenumber of samples to be processed in the dataset.

As in PCA, IncrementalPCA centers but does not scale theinput data for each feature before applying the SVD.

Examples:

Incremental PCA

2.5.1.3. PCA using randomized SVD

It is often interesting to project data to a lower-dimensionalspace that preserves most of the variance, by dropping the singular vectorof components associated with lower singular values.

For instance, if we work with 64x64 pixel gray-level picturesfor face recognition,the dimensionality of the data is 4096 and it is slow to train anRBF support vector machine on such wide data. Furthermore we know thatthe intrinsic dimensionality of the data is much lower than 4096 since allpictures of human faces look somewhat alike.The samples lie on a manifold of much lowerdimension (say around 200 for instance). The PCA algorithm can be usedto linearly transform the data while both reducing the dimensionalityand preserve most of the explained variance at the same time.

The class PCA used with the optional parametersvd_solver='randomized' is very useful in that case: since we are goingto drop most of the singular vectors it is much more efficient to limit thecomputation to an approximated estimate of the singular vectors we will keepto actually perform the transform.

For instance, the following shows 16 sample portraits (centered around0.0) from the Olivetti dataset. On the right hand side are the first 16singular vectors reshaped as portraits. Since we only require the top16 singular vectors of a dataset with size

and, the computation time isless than 1s:

If we note

and, the time complexityof the randomized PCA isinstead of for the exact methodimplemented 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 withsvd_solver='randomized' is not the exact inverse transform oftransform even when whiten=False (default).

Examples:

References:

“Finding structure with randomness: Stochastic algorithms forconstructing approximate matrix decompositions”Halko, et al., 2009

2.5.1.4. Kernel PCA

KernelPCA is an extension of PCA which achieves non-lineardimensionality reduction through the use of kernels (see Pairwise metrics, Affinities and Kernels). Ithas many applications including denoising, compression and structuredprediction (kernel dependency estimation). KernelPCA supports bothtransform and inverse_transform.

Examples:

Kernel PCA

2.5.1.5. Sparse principal components analysis (SparsePCA and MiniBatchSparsePCA)

SparsePCA is a variant of PCA, with the goal of extracting theset of sparse components that best reconstruct the data.

Mini-batch sparse PCA (MiniBatchSparsePCA) is a variant ofSparsePCA that is faster but less accurate. The increased speed isreached by iterating over small chunks of the set of features, for a givennumber of iterations.

Principal component analysis (PCA) has the disadvantage that thecomponents extracted by this method have exclusively dense expressions, i.e.they have non-zero coefficients when expressed as linear combinations of theoriginal variables. This can make interpretation difficult. In many cases,the real underlying components can be more naturally imagined as sparsevectors; for example in face recognition, components might naturally map toparts of faces.

Sparse principal components yields a more parsimonious, interpretablerepresentation, clearly emphasizing which of the original features contributeto the differences between samples.

The following example illustrates 16 components extracted using sparse PCA fromthe Olivetti faces dataset. It can be seen how the regularization term inducesmany zeros. Furthermore, the natural structure of the data causes the non-zerocoefficients to be vertically adjacent. The model does not enforce thismathematically: each component is a vector

, andthere is no notion of vertical adjacency except during the human-friendlyvisualization as 64x64 pixel images. The fact that the components shown belowappear local is the effect of the inherent structure of the data, which makessuch local patterns minimize reconstruction error. There exist sparsity-inducingnorms 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.

Note that there are many different formulations for the Sparse PCAproblem. The one implemented here is based on [Mrl09] . The optimizationproblem solved is a PCA problem (dictionary learning) with an

penalty on the components:

The sparsity-inducing

norm also prevents learningcomponents from noise when few training samples are available. The degreeof penalization (and thus sparsity) can be adjusted through thehyperparameter alpha. Small values lead to a gently regularizedfactorization, while larger values shrink many coefficients to zero.

Note

While in the spirit of an online algorithm, the classMiniBatchSparsePCA does not implement partial_fit becausethe algorithm is online along the features direction, not the samplesdirection.

Examples:

Faces dataset decompositions

References:

Mrl09
“Online Dictionary Learning for Sparse Coding”J. Mairal, F. Bach, J. Ponce, G. Sapiro, 2009
Jen09
“Structured Sparse Principal Component Analysis”R. Jenatton, G. Obozinski, F. Bach, 2009

2.5.2. 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 user-specified parameter.

When truncated SVD is applied to term-document matrices(as returned by CountVectorizer or TfidfVectorizer),this transformation is known aslatent semantic analysis(LSA), because it transforms such matricesto 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 sparseand 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 indexesfor information retrieval purposes.

Mathematically, truncated SVD applied to training samples

produces a low-rank 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) literatureswap the axes of the matrix

so that it has shapen_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.

TruncatedSVD is very similar to PCA, but differsin that it works on sample matrices

directlyinstead of their covariance matrices.When the columnwise (per-feature) means ofare subtracted from the feature values,truncated SVD on the resulting matrix is equivalent to PCA.In practical terms, this meansthat the TruncatedSVD transformer accepts scipy.sparsematrices without the need to densify them,as densifying may fill up memory even for medium-sized document collections.

While the TruncatedSVD transformerworks with any (sparse) feature matrix,using it on tf–idf matrices is recommended over raw frequency countsin an LSA/document processing setting.In particular, sublinear scaling and inverse document frequencyshould 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.

Examples:

Clustering text documents using k-means

References:

Christopher D. Manning, Prabhakar Raghavan and Hinrich Schütze (2008),Introduction to Information Retrieval, Cambridge University Press,chapter 18: Matrix decompositions & latent semantic indexing

2.5.3. Dictionary Learning

2.5.3.1. Sparse coding with a precomputed dictionary

The SparseCoder object is an estimator that can be used to transform signalsinto sparse linear combination of atoms from a fixed, precomputed dictionarysuch as a discrete wavelet basis. This object therefore does notimplement a fit method. The transformation amountsto a sparse coding problem: finding a representation of the data as a linearcombination of as few dictionary atoms as possible. All variations ofdictionary learning implement the following transform methods, controllable viathe transform_method initialization parameter:

Orthogonal matching pursuit (Orthogonal Matching Pursuit (OMP))
Least-angle regression (Least Angle Regression)
Lasso computed by least-angle 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 imagereconstruction tasks, orthogonal matching pursuit yields the most accurate,unbiased reconstruction.

The dictionary learning objects offer, via the split_code parameter, thepossibility to separate the positive and negative values in the results ofsparse coding. This is useful when dictionary learning is used for extractingfeatures that will be used for supervised learning, because it allows thelearning algorithm to assign different weights to negative loadings of aparticular atom, from to the corresponding positive loading.

The split code for a single sample has length 2 * n_componentsand is constructed using the following rule: First, the regular code of lengthn_components is computed. Then, the first n_components entries of thesplit_code arefilled with the positive part of the regular code vector. The second half ofthe split code is filled with the negative part of the code vector, only witha positive sign. Therefore, the split_code is non-negative.

Examples:

Sparse coding with a precomputed dictionary

2.5.3.2. Generic dictionary learning

Dictionary learning (DictionaryLearning) is a matrix factorizationproblem that amounts to finding a (usually overcomplete) dictionary that willperform well at sparsely encoding the fitted data.

Representing data as sparse combinations of atoms from an overcompletedictionary is suggested to be the way the mammalian primary visual cortex works.Consequently, dictionary learning applied on image patches has been shown togive 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 updatingthe sparse code, as a solution to multiple Lasso problems, considering thedictionary fixed, and then updating the dictionary to best fit the sparse code.

After using such a procedure to fit the dictionary, the transform is simply asparse coding step that shares the same implementation with all dictionarylearning objects (see Sparse coding with a precomputed dictionary).

It is also possible to constrain the dictionary and/or code to be positive tomatch constraints that may be present in the data. Below are the faces withdifferent positivity constraints applied. Red indicates negative values, blueindicates positive values, and white represents zeros.

The following image shows how a dictionary learned from 4x4 pixel image patchesextracted from part of the image of a raccoon face looks like.

Examples:

Image denoising using dictionary learning

References:

“Online dictionary learning for sparse coding”J. Mairal, F. Bach, J. Ponce, G. Sapiro, 2009

2.5.3.3. Mini-batch dictionary learning

MiniBatchDictionaryLearning implements a faster, but less accurateversion of the dictionary learning algorithm that is better suited for largedatasets.

By default, MiniBatchDictionaryLearning divides the data intomini-batches and optimizes in an online manner by cycling over the mini-batchesfor the specified number of iterations. However, at the moment it does notimplement a stopping condition.

The estimator also implements partial_fit, which updates the dictionary byiterating only once over a mini-batch. This can be used for online learningwhen the data is not readily available from the start, or for when the datadoes 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 thedictionary. For instance the MiniBatchKMeans estimator iscomputationally efficient and implements on-line learning with apartial_fit method.

Example: Online learning of a dictionary of parts of faces

2.5.4. Factor Analysis

In unsupervised learning we only have a dataset

. How can this dataset be described mathematically? A very simplecontinuous latent variable model for is

The vector

is called “latent” because it is unobserved. isconsidered a noise term distributed according to a Gaussian with mean 0 andcovariance (i.e.), is somearbitrary offset vector. Such a model is called “generative” as it describeshow is generated from. If we use all the’s as columns to forma matrix and all the’s as columns of a matrixthen we can write (with suitably defined and):

In other words, we decomposed matrix

is given, the above equation automatically implies the followingprobabilistic interpretation:

For a complete probabilistic model we also need a prior distribution for thelatent variable

. The most straightforward assumption (based on the niceproperties 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 meanand a covariance. We need to impose some more specific structure on oneof these two parameters. A simple additional assumption regards thestructure of the error covariance:

: This assumption leads tothe probabilistic model of PCA.

: This model is calledFactorAnalysis, a classical statistical model. The matrix W issometimes 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 complexmodels, e.g. Mixture of Factor Analysers. One gets very different models (e.g.FastICA) if non-Gaussian priors on the latent variables are assumed.

Factor analysis can produce similar components (the columns of its loadingmatrix) to PCA. However, one can not make any general statementsabout these components (e.g. whether they are orthogonal):

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

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

Examples:

Model selection with Probabilistic PCA and Factor Analysis (FA)

2.5.5. Independent component analysis (ICA)

Independent component analysis separates a multivariate signal intoadditive subcomponents that are maximally independent. It isimplemented in scikit-learn using the Fast ICAalgorithm. Typically, ICA is not used for reducing dimensionality butfor separating superimposed signals. Since the ICA model does not includea noise term, for the model to be correct, whitening must be applied.This can be done internally using the whiten argument or manually using oneof the PCA variants.

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

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

Examples:

2.5.6. Non-negative matrix factorization (NMF or NNMF)

2.5.6.1. NMF with the Frobenius norm

NMF1 is an alternative approach to decomposition that assumes that thedata and the components are non-negative. NMF can be plugged ininstead of PCA or its variants, in the cases where the data matrixdoes not contain negative values. It finds a decomposition of samples

into two matrices and of non-negative elements,by optimizing the distance between and the matrix product. The most widely used distance function is the squared Frobeniusnorm, which is an obvious extension of the Euclidean norm to matrices:

Unlike PCA, the representation of a vector is obtained in an additivefashion, by superimposing the components, without subtracting. Such additivemodels are efficient for representing images and text.

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

The init attribute determines the initialization method applied, whichhas a great impact on the performance of the method. NMF implements themethod Nonnegative Double Singular Value Decomposition. NNDSVD 4 is based ontwo SVD processes, one approximating the data matrix, the other approximatingpositive sections of the resulting partial SVD factors utilizing an algebraicproperty of unit rank matrices. The basic NNDSVD algorithm is better fit forsparse factorization. Its variants NNDSVDa (in which all zeros are set equal tothe mean of all elements of the data), and NNDSVDar (in which the zeros are setto random perturbations less than the mean of the data divided by 100) arerecommended in the dense case.

Note that the Multiplicative Update (‘mu’) solver cannot update zeros present inthe initialization, so it leads to poorer results when used jointly with thebasic NNDSVD algorithm which introduces a lot of zeros; in this case, NNDSVDa orNNDSVDar should be preferred.

NMF can also be initialized with correctly scaled random non-negativematrices by setting init="random". An integer seed or aRandomState can also be passed to random_state to controlreproducibility.

In NMF, L1 and L2 priors can be added to the loss function in orderto regularize the model. The L2 prior uses the Frobenius norm, while the L1prior uses an elementwise L1 norm. As in ElasticNet, we control thecombination 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 functionnon_negative_factorization allows a finer control through theregularization attribute, and may regularize only W, only H, or both.

2.5.6.2. NMF with a beta-divergence

As described previously, the most widely used distance function is the squaredFrobenius norm, which is an obvious extension of the Euclidean norm tomatrices:

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

Or, the Itakura-Saito (IS) divergence:

These three distances are special cases of the beta-divergence family, with

respectively 6. The beta-divergence aredefined by :

Note that this definition is not valid if

, yet it canbe continuously extended to the definitions of andrespectively.

NMF implements two solvers, using Coordinate Descent (‘cd’) 5, andMultiplicative Update (‘mu’) 6. The ‘mu’ solver can optimize everybeta-divergence, including of course the Frobenius norm (

), the(generalized) Kullback-Leibler divergence () and theItakura-Saito divergence (). Note that for, the ‘mu’ solver is significantly faster than for othervalues of. Note also that with a negative (or 0, i.e.‘itakura-saito’), the input matrix cannot contain zero values.

The ‘cd’ solver can only optimize the Frobenius norm. Due to theunderlying non-convexity of NMF, the different solvers may converge todifferent minima, even when optimizing the same distance function.

NMF is best used with the fittransform 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 thesestored 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)

Examples:

References:

1
“Learning the parts of objects by non-negative matrix factorization”D. Lee, S. Seung, 1999
2
“Non-negative Matrix Factorization with Sparseness Constraints”P. Hoyer, 2004
4
“SVD based initialization: A head start for nonnegativematrix factorization”C. Boutsidis, E. Gallopoulos, 2008
5
“Fast local algorithms for large scale nonnegative matrix and tensorfactorizations.”A. Cichocki, A. Phan, 2009
6(1,2)
“Algorithms for nonnegative matrix factorization with the beta-divergence”C. Fevotte, J. Idier, 2011

2.5.7. Latent Dirichlet Allocation (LDA)

Latent Dirichlet Allocation is a generative probabilistic model for collections ofdiscrete dataset such as text corpora. It is also a topic model that is used fordiscovering abstract topics from a collection of documents.

The graphical model of LDA is a three-level generative model:Note on notations presented in the graphical model above, which can be found inHoffman 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 thegenerative process. A shaded node indicates an observed variable and an unshadednode indicates a hidden (latent) variable. In this case, words in the corpus arethe only data that we observe. The latent variables determine the random mixtureof 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 topicstructure.

When modeling text corpora, the model assumes the following generative processfor a corpus with

documents and topics, withcorresponding 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 methoduses a simpler distribution

to approximate it, and those variational parameters,, are optimized to maximize the EvidenceLower Bound (ELBO):

Maximizing ELBO is equivalent to minimizing the Kullback-Leibler(KL) divergencebetween

and the true posterior.

LatentDirichletAllocation implements the online variational Bayesalgorithm and supports both online and batch update methods.While the batch method updates variational variables after each full pass throughthe data, the online method updates variational variables from mini-batch datapoints.

Note

Although the online method is guaranteed to converge to a local optimum point, the quality ofthe optimum point and the speed of convergence may depend on mini-batch size andattributes related to learning rate setting.

When LatentDirichletAllocation is applied on a “document-term” matrix, the matrixwill 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” matrixcan be calculated from transform method.

LatentDirichletAllocation also implements partial_fit method. This is usedwhen data can be fetched sequentially.

Examples:

Topic extraction with Non-negative Matrix Factorization and Latent Dirichlet Allocation

References:

“Latent Dirichlet Allocation”D. Blei, A. Ng, M. Jordan, 2003
“Online Learning for Latent Dirichlet Allocation”M. Hoffman, D. Blei, F. Bach, 2010
“Stochastic Variational Inference”M. Hoffman, D. Blei, C. Wang, J. Paisley, 2013

See also Dimensionality reduction for dimensionality reduction withNeighborhood Components Analysis.