Sparse PCA

I have been working on the integration into the scikits.learn codebase of a sparse principal components analysis (SparsePCA) algorithm coded by Gaël and Alexandre and based on [[1]][]. Because the name “sparse PCA” has some inherent ambiguity, I will describe in greater depth what problem we are actually solving, and what it can be used for.

The problem

Mathematically, this implementation of Sparse PCA solves:

\$latex (U\^*, V\^*)=\underset{U,V}{\mathrm{argmin\,}}\frac{1}{2}||X-UV||_2\^2+\alpha||V||_1\$

with \$latex || U_k ||_2 = 1\$ for all \$latex 0 \leq k \< n_{atoms}\$

This looks really abstract so let’s try to interpret it. We are looking for a matrix factorization \$latex UV\$ of \$latex X \in \mathbf{R}\^{n_{samples}\times n_{features}}\$, just like in ordinary PCA. The interpretation is that the \$latex n_{atoms}\$ lines of \$latex V\$ are the extracted components, while the lines of \$latex U\$ are the coordinates of the samples in this projection.

The most important difference between this and PCA is that we enforce sparsity on the components. In other words, we look for a representation of the data as a linear combination of sparse signals.

Another difference is that, unlike in PCA, here we don’t constrain U to be orthogonal, just to consist of normalized column vectors. There are different approaches where this constraint appears too, and they are on the list for this summer, but I digress.

The approach

As usual, such optimization problems are solved by alternatively minimizing one of the variables while keeping the other fixed, until convergence is reached.

The update of \$latex V\$ (the dictionary) is computed as the solution of a Lasso least squares problem.  We allow the user to choose between the least angle regression method (LARS) or stochastic gradient descent as algorithms to solve the Lasso problem.

The update of \$latex U\$ is block coordinate descent with warm restart. This is a batch adaptation of an online algorithm proposed by Mairal et al. in [[1]][].

Sparse PCA as a transformer

Of course, in order to be of practical use, the code needs to be refactored into a scikits.learn transformer object, just like scikits.learn.decomposition.pca. This means that the optimization problem described above corresponds to the fitting stage. The post-fit state of the transformer is given by the learned components (the matrix \$latex V\$ above).

In order to transform new data according to the learned sparse PCA model (for example, prior to classification of the test data), we simply need to do a least squares projection of the new data on the sparse components.

What is it good for?

For applications such as text and image processing, its great advantage is interpretability. When running a regular PCA on a set of documents in bag of words format, we can find an interesting visualisation on a couple of components, and it can show discrimination or clusters. The biggest problem is that the maximum variance components found by PCA have very dense expressions as linear combinations of the initial features. In practice, sometimes interpretation is made by simply marking the \$latex k\$ variables with the highest coefficients in this representation, and basically interpreting as if the rest are truncated to 0 (this has been taught to me in a class on PCA interpretation).

Such an approximation can be highly misleading, and now we offer you the sparse PCA code that can extract components with only few non-zero coefficients, and therefore easy to interpret.

For image data, sparse PCA should extract local components such as, famously, parts of the face in the case of face recognition.

Personally I can’t wait to have it ready for the scikit so that I can play with it in some of my projects. I have two tasks where I can’t wait to see the results: one is related to Romanian infinitives, where PCA revealed structure, and I would love to see how it looks with sparse n-gram components. The other task is to plug it in as feature extractor for handwritten digit classification, for my undergraduate thesis.


[[1]]: #footnote_1

Comments !