Distributed compressed sensing algorithms: Completing the puzzle

Reconstructing compressed sensing signals involves solving an optimization problem. An example is Basis Pursuit (BP) [1], which is applicable only in noise-free scenarios. In noisy scenarios, either the Basis Pursuit Denoising (BPDN) [1] or the Noise-Aware BP (NABP) [2] can be used. Consider a distributed scenario where the dictionary matrix and the vector of observations are spread over the nodes of a network. We solve the following open problem: design distributed algorithms that solve BPDN with a column partition, i.e., when each node knows only some columns of the dictionary matrix, and that solve NABP with a row partition, i.e., when each node knows only some rows of the dictionary matrix and the corresponding observations. Our approach manipulates these problems so that a recent general-purpose algorithm for distributed optimization can be applied.