Insights into the stable recovery of sparse solutions in overcomplete representations using network information theory

In this paper, we examine the problem of overcomplete representations and provide new insights into the problem of stable recovery of sparse solutions in noisy environments. We establish an important connection between the inverse problem that arises in overcomplete representations and wireless communication models in network information theory. We show that the stable recovery of a sparse solution with a single measurement vector (SMV) can be viewed as decoding competing users simultaneously transmitting messages through a Multiple Access Channel (MAC) at the same rate. With multiple measurement vectors (MMV), we relate the inverse problem to the wireless communication scenario with a Multiple-Input Multiple-Output (MIMO) channel. In each case, based on the connection established between the two domains, we leverage channel capacity results with outage analysis to shed light on the fundamental limits of any algorithm to stably recover sparse solutions in the presence of noise. Our results explicitly indicate the conditions on the key model parameters, e.g. degree of overcompleteness, degree of sparsity, and the signal-to-noise ratio, to guarantee the existence of asymptotically stable reconstruction of the sparse source.