Closed-form EM for sparse coding and its application to source separation

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 poten-tially multi-modal posterior into account. The computational cost of the algo-rithm scales exponentially with the number of hidden dimensions. However, with current computational resources, it is still possible to efficiently learn model pa-rameters 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 competi-tive with other recent methods