On rank awareness, thresholding, and MUSIC for joint sparse recovery

Abstract

This letter establishes sufficient conditions for the sparse multiple measurement vector (MMV) or row-sparse matrix approximation problem for the Rank Aware Row Thresholding (RART) algorithm. Using the rank aware selection operator to define RART results in discrete MUltiple SIgnal Classification (MUSIC) from array signal processing. When the sensing matrix is drawn from the random Gaussian matrix ensemble, we establish that the rank of the row-sparse measurement matrix in the noiseless row-sparse recovery problem allows RART (MUSIC) to reduce the effect of the log ⁡ ( n ) penalty term that is present in traditional compressed sensing results and simultaneously provides a row-rank deficient recovery result for MUSIC. Empirical evidence shows that Thresholding closely matches RART in successful row-sparse approximation. The theoretical and empirical evidence provides further support for the conjecture that the thresholding operator in more sophisticated greedy algorithms is the source of their observed rank awareness.