On the role of the properties of the nonzero entries on sparse signal recovery

We study the role of the nonzero entries on the performance limits in support recovery of sparse signals. The key to our results is the recently studied connection between sparse signal recovery and multiple user communication. By leveraging the concept of outage capacity in information theory, we explicitly characterize the impact of the probability distribution imposed on the nonzero entries of the sparse signal on support recovery. When Multiple Measurement Vectors (MMV) are available, we show that the identification of the nonzero rows of the signal is closely connected to decoding the messages from multiple users over a Single-Input Multiple-Output channel. Necessary and sufficient conditions for support (indices of nonzero rows) recovery are provided, and the results allow us to understand the role of correlation of the nonzero entries as well as the role of the rank of the matrix formed from the non-zero entries.