Support Recovery of Sparse Signals in the Presence of Multiple Measurement Vectors

TL;DR

This paper establishes theoretical conditions for accurately recovering the support of sparse signals using multiple measurement vectors, linking the problem to communication theory and highlighting the benefits of multiple measurements.

Contribution

It introduces an information theoretic framework for MMV support recovery, deriving sharp conditions and emphasizing the role of matrix rank in performance limits.

Findings

Support recovery conditions depend on measurements, sparsity, noise, and number of vectors.

MMV improves support recovery performance compared to single measurement scenarios.

Matrix rank of nonzero entries significantly influences recovery success.

Abstract

This paper studies the problem of support recovery of sparse signals based on multiple measurement vectors (MMV). The MMV support recovery problem is connected to the problem of decoding messages in a Single-Input Multiple-Output (SIMO) multiple access channel (MAC), thereby enabling an information theoretic framework for analyzing performance limits in recovering the support of sparse signals. Sharp sufficient and necessary conditions for successful support recovery are derived in terms of the number of measurements per measurement vector, the number of nonzero rows, the measurement noise level, and especially the number of measurement vectors. Through the interpretations of the results, in particular the connection to the multiple output communication system, the benefit of having MMV for sparse signal recovery is illustrated providing a theoretical foundation to the performance improvement enabled by MMV as observed in many existing simulation results. In particular, it is shown that the structure (rank) of the matrix formed by the nonzero entries plays an important role on the performance limits of support recovery.