A Noniterative Online Bayesian Algorithm for the Recovery of Temporally Correlated Sparse Vectors

In this paper, we address the problem of online (sequential) recovery of temporally correlated sparse vectors sharing a common support, from noisy underdetermined linear measurements. The temporal correlation of the sparse vectors is modeled using a first-order autoregressive process. The online algorithm is formulated using the sparse Bayesian learning framework and is implemented using a sequential expectation-maximization procedure. Our algorithm is noniterative in nature, and requires less computational and memory resources compared to offline processing. We analyze the convergence of the algorithm in the case when the sparse vectors are uncorrelated, using tools from stochastic approximation theory. We show that the sequence of the covariance estimates converge either to the global minimum of the offline equivalent cost function or to the all zero vector, regardless of the sparsity level of the signal. Through numerical results, we demonstrate the efficacy of the proposed online algorithm and compare it with its offline counterpart as well as with existing online sparse vector recovery algorithms. We also illustrate the performance of the algorithm in the context of sparse orthogonal frequency division multiplexing channel estimation.