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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 successive orthogonal components that explain a maximum amount of the variance. In scikit-learn, PCA is implemented as a transformer object that learns n components in its fit method, and can be used on new data to project it on these components.

The optional parameter whiten=True parameter make 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. However in that case the inverse transform is no longer exact since some information is lost while forward transforming.

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

../_images/plot_pca_vs_lda_0011.png

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 cross-validation:

../_images/plot_pca_vs_fa_model_selection_0011.png

2.5.1.2. Approximate PCA

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

orig_img pca_img

RandomizedPCA can hence be used as a drop in replacement for PCA with the exception that we need to give it the size of the lower-dimensional space n_components as a mandatory input parameter.

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

The memory footprint of RandomizedPCA is also proportional to 2 \cdot n_{max} \cdot n_{components} instead of n_{max}
\cdot n_{min} for the exact method.

Furthermore RandomizedPCA is able to work with scipy.sparse matrices as input which make it suitable for reducing the dimensionality of features extracted from text documents for instance.

Note: the implementation of inverse_transform in RandomizedPCA is not the exact inverse transform of transform even when whiten=False (default).

2.5.1.3. Kernel PCA

KernelPCA is an extension of PCA which achieves non-linear dimensionality reduction through the use of kernels. It has many applications including denoising, compression and structured prediction (kernel dependency estimation). KernelPCA supports both transform and inverse_transform.

../_images/plot_kernel_pca_0011.png

Examples:

2.5.1.4. 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.

Mini-batch 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 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 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 see [Jen09] for a review of such methods. For more details on how to use Sparse PCA, see the Examples section below.

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 \ell_1 penalty on the components:

(U^*, V^*) = \underset{U, V}{\operatorname{arg\,min\,}} & \frac{1}{2}
             ||X-UV||_2^2+\alpha||V||_1 \\
             \text{subject to\,} & ||U_k||_2 = 1 \text{ for all }
             0 \leq k < n_{components}

The sparsity-inducing \ell_1 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 implement partial_fit because the algorithm is online along the features direction, not the samples direction.

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 k largest singular values, where k is a user-specified parameter.

When truncated SVD is applied to term-document 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 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 X produces a low-rank approximation X:

X \approx X_k = U_k \Sigma_k V_k^\top

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

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

X' = X V_k^\top

Note

Most treatments of LSA in the natural language processing (NLP) and information retrieval (IR) literature swap the axis of the matrix 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.

TruncatedSVD is very similar to PCA, but differs in that it works on sample matrices X directly instead of their covariance matrices. When the columnwise (per-feature) means of X 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 medium-sized 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.

References:

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 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:

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.

2.5.3.2. Generic dictionary learning

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

Representing data as sparse combinations of atoms from an overcomplete dictionary is suggested to be the way the mammal 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.

(U^*, V^*) = \underset{U, V}{\operatorname{arg\,min\,}} & \frac{1}{2}
             ||X-UV||_2^2+\alpha||U||_1 \\
             \text{subject to\,} & ||V_k||_2 = 1 \text{ for all }
             0 \leq k < n_{atoms}

pca_img2 dict_img2

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).

The following image shows how a dictionary learned from 4x4 pixel image patches extracted from part of the image of Lena looks like.

../_images/plot_image_denoising_0011.png

References:

2.5.3.3. Mini-batch 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 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.

../_images/plot_dict_face_patches_0011.png

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 on-line learning partial_fit method.

2.5.4. Factor Analysis

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

x_i = W h_i + \mu + \epsilon

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

\mathbf{X} = W \mathbf{H} + \mathbf{M} + \mathbf{E}

In other words, we decomposed matrix \mathbf{X}.

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

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 h. The most straightforward assumption (based on the nice properties of the Gaussian distribution) is h \sim \mathcal{N}(0,
\mathbf{I}). This yields a Gaussian as the marginal distribution of x:

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

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

  • \Psi = \sigma^2 \mathbf{I}: This assumption leads to the probabilistic model of PCA.
  • \Psi = diag(\psi_1, \psi_2, \dots, \psi_n): This model is called Factor Analysis, a classical statistical model. The matrix W is sometimes called factor loading matrix.

Both model 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. FastICA) if non-Gaussian 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):

pca_img3 fa_img3

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):

../_images/plot_faces_decomposition_0081.png

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

../_images/plot_pca_vs_fa_model_selection_0021.png

2.5.5. 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 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:

../_images/plot_ica_blind_source_separation_0011.png

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

pca_img4 ica_img4

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

NMF is an alternative approach to decomposition that assumes that the data and the components are non-negative. 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 X into two matrices V and H of non-negative elements, by optimizing for the squared Frobenius norm:

\arg\min_{W,H} ||X - WH||^2 = \sum_{i,j} X_{ij} - {WH}_{ij}

This norm is an obvious extension of the Euclidean norm to matrices. (Other optimization objectives have been suggested in the NMF literature, in particular Kullback-Leibler divergence, but these are not currently implemented.)

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, 04] that, when carefully constrained, NMF can produce a parts-based 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.

pca_img5 nmf_img5

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.

NMF can also be initialized with random non-negative matrices, by passing an integer seed or a RandomState to init.

In NMF, sparseness can be enforced by setting the attribute sparseness to "data" or "components". Sparse components lead to localized features, and sparse data leads to a more efficient representation of the data.

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