2.5. Decomposing signals in components (matrix factorization problems) (2024)

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 \(n\) 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

• PCA example with Iris Data-set

• Comparison of LDA and PCA 2D projection of Iris dataset

• Model selection with Probabilistic PCA and Factor Analysis (FA)

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 memory mapped file usingnumpy.memmap.

IncrementalPCA only stores estimates of component and noise variances,in order update explained_variance_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 \(n_{samples} = 400\)and \(n_{features} = 64 \times 64 = 4096\), the computation time isless than 1s:

If we note \(n_{\max} = \max(n_{\mathrm{samples}}, n_{\mathrm{features}})\) and\(n_{\min} = \min(n_{\mathrm{samples}}, n_{\mathrm{features}})\), the time complexityof the randomized PCA is \(O(n_{\max}^2 \cdot n_{\mathrm{components}})\)instead of \(O(n_{\max}^2 \cdot n_{\min})\) for the exact methodimplemented in PCA.

The memory footprint of randomized PCA is also proportional to\(2 \cdot n_{\max} \cdot n_{\mathrm{components}}\) instead of \(n_{\max}\cdot n_{\min}\) 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

• Faces recognition example using eigenfaces and SVMs

• Faces dataset decompositions

References

• Algorithm 4.3 in“Finding structure with randomness: Stochastic algorithms forconstructing approximate matrix decompositions”Halko, et al., 2009

• “An implementation of a randomized algorithm for principal componentanalysis” A. Szlam et al. 2014

2.5.1.4. 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 \(h \in \mathbf{R}^{4096}\), 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\(\ell_1\) penalty on the components:

\[\begin{split}(U^*, V^*) = \underset{U, V}{\operatorname{arg\,min\,}} & \frac{1}{2} ||X-UV||_{\text{Fro}}^2+\alpha||V||_{1,1} \\ \text{subject to } & ||U_k||_2 <= 1 \text{ for all } 0 \leq k < n_{components}\end{split}\]

\(||.||_{\text{Fro}}\) stands for the Frobenius norm and \(||.||_{1,1}\)stands for the entry-wise matrix norm which is the sum of the absolute valuesof all the entries in the matrix.The sparsity-inducing \(||.||_{1,1}\) matrix 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.

Examples

• Faces dataset decompositions

References

2.5.2.1. Exact Kernel PCA#

KernelPCA is an extension of PCA which achieves non-lineardimensionality reduction through the use of kernels (see Pairwise metrics, Affinities and Kernels) [Scholkopf1997]. Ithas many applications including denoising, compression and structuredprediction (kernel dependency estimation). KernelPCA supports bothtransform and inverse_transform.

Examples

• Kernel PCA

  See Also

  3. An Informal Introduction to Python1.5. Stochastic Gradient Descentrandom — Generate pseudo-random numbersPPLATO | FLAP | PHYS 2.3: Forces

• Image denoising using kernel PCA

References

2.5.2.2. Choice of solver for Kernel PCA#

While in PCA the number of components is bounded by the number offeatures, in KernelPCA the number of components is bounded by thenumber of samples. Many real-world datasets have large number of samples! Inthese cases finding all the components with a full kPCA is a waste ofcomputation time, as data is mostly described by the first few components(e.g. n_components<=100). In other words, the centered Gram matrix thatis eigendecomposed in the Kernel PCA fitting process has an effective rank thatis much smaller than its size. This is a situation where approximateeigensolvers can provide speedup with very low precision loss.

2.5.4.1. 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

2.5.4.2. 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.

\[\begin{split}(U^*, V^*) = \underset{U, V}{\operatorname{arg\,min\,}} & \frac{1}{2} ||X-UV||_{\text{Fro}}^2+\alpha||U||_{1,1} \\ \text{subject to } & ||V_k||_2 <= 1 \text{ for all } 0 \leq k < n_{\mathrm{atoms}}\end{split}\]

\(||.||_{\text{Fro}}\) stands for the Frobenius norm and \(||.||_{1,1}\)stands for the entry-wise matrix norm which is the sum of the absolute valuesof all the entries in the matrix.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

2.5.4.3. 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.

2.5.7.1. NMF with the Frobenius norm#

NMF [1] 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\(X\) into two matrices \(W\) and \(H\) of non-negative elements,by optimizing the distance \(d\) between \(X\) and the matrix product\(WH\). The most widely used distance function is the squared Frobeniusnorm, which is an obvious extension of the Euclidean norm to matrices:

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

Unlike 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 order toregularize the model. The L2 prior uses the Frobenius norm, while the L1 prioruses an elementwise L1 norm. As in ElasticNet,we control the combination of L1 and L2 with the l1_ratio (\(\rho\))parameter, and the intensity of the regularization with the alpha_W andalpha_H (\(\alpha_W\) and \(\alpha_H\)) parameters. The priors arescaled by the number of samples (\(n\_samples\)) for H and the number offeatures (\(n\_features\)) for W to keep their impact balanced withrespect to one another and to the data fit term as independent as possible ofthe size of the training set. Then the priors terms are:

\[(\alpha_W \rho ||W||_1 + \frac{\alpha_W(1-\rho)}{2} ||W||_{\mathrm{Fro}} ^ 2) * n\_features+ (\alpha_H \rho ||H||_1 + \frac{\alpha_H(1-\rho)}{2} ||H||_{\mathrm{Fro}} ^ 2) * n\_samples\]

and the regularized objective function is:

\[d_{\mathrm{Fro}}(X, WH)+ (\alpha_W \rho ||W||_1 + \frac{\alpha_W(1-\rho)}{2} ||W||_{\mathrm{Fro}} ^ 2) * n\_features+ (\alpha_H \rho ||H||_1 + \frac{\alpha_H(1-\rho)}{2} ||H||_{\mathrm{Fro}} ^ 2) * n\_samples\]

2.5.7.2. 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:

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

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

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

Or, the Itakura-Saito (IS) divergence:

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

These three distances are special cases of the beta-divergence family, with\(\beta = 2, 1, 0\) respectively [6]. The beta-divergence aredefined by :

\[d_{\beta}(X, Y) = \sum_{i,j} \frac{1}{\beta(\beta - 1)}(X_{ij}^\beta + (\beta-1)Y_{ij}^\beta - \beta X_{ij} Y_{ij}^{\beta - 1})\]

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

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 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

• Faces dataset decompositions

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

2.5.7.3. Mini-batch Non Negative Matrix Factorization#

MiniBatchNMF [7] implements a faster, but less accurate version of thenon negative matrix factorization (i.e. NMF),better suited for large datasets.

By default, MiniBatchNMF divides the data into mini-batches andoptimizes the NMF model in an online manner by cycling over the mini-batchesfor the specified number of iterations. The batch_size parameter controlsthe size of the batches.

In order to speed up the mini-batch algorithm it is also possible to scalepast batches, giving them less importance than newer batches. This is doneintroducing a so-called forgetting factor controlled by the forget_factorparameter.

The estimator also implements partial_fit, which updates H by iteratingonly once over a mini-batch. This can be used for online learning when the datais not readily available from the start, or when the data does not fit into memory.

References