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
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:
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:
Examples:
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 out-of-core Principal Component Analysis either by:
partial_fit
method on chunks of data fetched sequentially from the local hard drive or a network database.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.
Examples:
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 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 \(n_{samples} = 400\) and \(n_{features} = 64 \times 64 = 4096\), 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 lower-dimensional space n_components
as a mandatory input parameter.
If we note \(n_{\max} = \max(n_{\mathrm{samples}}, n_{\mathrm{features}})\) and \(n_{\min} = \min(n_{\mathrm{samples}}, n_{\mathrm{features}})\), the time complexity of the randomized PCA
is \(O(n_{\max}^2 \cdot n_{\mathrm{components}})\) instead of \(O(n_{\max}^2 \cdot n_{\min})\) for the exact method implemented 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
with svd_solver='randomized'
is not the exact inverse transform of transform
even when whiten=False
(default).
References:
KernelPCA
is an extension of PCA which achieves non-linear 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
.
Examples:
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 [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 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:
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.
Examples:
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 |
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 \(X\):
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 \(V_k\):
Note
Most treatments of LSA in the natural language processing (NLP) and information retrieval (IR) literature swap the axes 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.
Examples:
References:
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.
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.
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.
References:
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.
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 with a partial_fit
method.
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 variable
model for \(X\) is
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 \(\mathbf{E}\)):
In other words, we decomposed matrix \(\mathbf{X}\).
If \(h_i\) 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 \(h\). 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 \(x\):
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 \(\Psi\):
PCA
.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. 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):
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 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:
ICA can also be used as yet another non linear decomposition that finds components with some sparsity:
Examples:
NMF
[1] 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 \(W\) 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 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] [2] 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.