Bayesian Simultaneous Sparse Approximation With Smooth Signals

In the simultaneous sparse approximation problem, several latent vectors corresponding to independent random realizations from a common sparsity profile are recovered from an undercomplete set of measurements. In this paper, we address an extension of this problem, where in addition to the common sparsity profile, the vectors of interest are assumed to have a high correlation among each other. Specifically, we consider the case when the non-zero rows in the combined latent signal vectors are considered to be temporally smooth signals. We present a Bayesian formulation of the problem and develop a greedy inference algorithm based on sparse Bayesian learning for independent observations. A difficulty is that unlike for existing greedy methods, there is no closed form expression for the maximizer of the objective function in the greedy algorithm when row correlations are introduced. We derive two methods to maximize the objective function, one based on the EM algorithm and another on a fixed-point iteration. Empirical results show that the proposed method provides better reconstruction results compared to existing methods, especially when the signal-to-noise ratio is low and the latent signal vectors are highly correlated. We also demonstrate the application of the proposed method to source localization in magnetoencephalography, where it obtains a temporally smooth solution with accurate localization of the brain activity.