RECOVERING JOINTLY SPARSE VECTORS VIA HARD THRESHOLDING PURSUIT

We introduce an iterative algorithm designed to find row-sparse matrices X ∈ RN×K solution of an underdetermined linear system AZ = Y , where A ∈ Rm×N and Y ∈ Rm×K are given. In the case K = 1, the algorithm is a simple combination of popular Compressive Sensing algorithms, which we had previously coined Hard Thresholding Pursuit. After recalling the main results concerning this algorithm, we generalize them to the case K ≥ 1 for the new Simultaneous Hard Thresholding Pursuit algorithm. In particular, we prove that any s-row-sparse matrix can be exactly recovered using a finite number of iterations of the algorithm provided that the 3sth Restricted Isometry Constant of the matrix A satisfies δ3s < 1/ √ 3. We also discuss the cost of recovering matrices at once via Simultaneous Hard Thresholding Pursuit versus recovering their columns one by one via Hard Thresholding Pursuit. Keywords— compressive sensing, sparse recovery, iterative algorithms, thresholding, joint sparsity