Closed-form EM for Sparse Coding and its Application to Source Separation

TL;DR

This paper introduces a novel sparse coding algorithm based on closed-form EM updates with continuous latent variables, capable of handling multi-modal posteriors and applicable to source separation tasks.

Contribution

It presents the first closed-form EM algorithm for sparse coding using a spike-and-slab prior, extending probabilistic PCA and enabling efficient learning for medium-scale problems.

Findings

Verifies likelihood maximization on artificial data

Recovers sparse directions effectively

Performs competitively on real source separation benchmarks

Abstract

We define and discuss the first sparse coding algorithm based on closed-form EM updates and continuous latent variables. The underlying generative model consists of a standard `spike-and-slab' prior and a Gaussian noise model. Closed-form solutions for E- and M-step equations are derived by generalizing probabilistic PCA. The resulting EM algorithm can take all modes of a potentially multi-modal posterior into account. The computational cost of the algorithm scales exponentially with the number of hidden dimensions. However, with current computational resources, it is still possible to efficiently learn model parameters for medium-scale problems. Thus the model can be applied to the typical range of source separation tasks. In numerical experiments on artificial data we verify likelihood maximization and show that the derived algorithm recovers the sparse directions of standard sparse coding distributions. On source separation benchmarks comprised of realistic data we show that the algorithm is competitive with other recent methods.