Joint Sparse Recovery Using Signal Space Matching Pursuit
Junhan Kim, Jian Wang, Luong Trung Nguyen, Byonghyo Shim
Abstract
In this paper, we put forth a new joint sparse recovery algorithm called signal space matching pursuit (SSMP). The key idea of the proposed SSMP algorithm is to sequentially investigate the support of jointly sparse vectors to minimize the subspace distance to the residual space. Our performance guarantee analysis indicates that SSMP accurately reconstructs any row K-sparse matrix of rank r in the full row rank scenario if the sampling matrix A satisfies krank(A) ≥ K+1, which meets the fundamental minimum requirement on A to ensure exact recovery. We also show that SSMP guarantees exact reconstruction in at most K - r + [r/L] iterations, provided that A satisfies the restricted isometry property (RIP) of order L(K - r) + r + 1 %/L with δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L(K-r)+r+1</sub> <; max{ √r/K+ r/4 + √r/4, √L/K +1.15√L), where, L is the number of indices chosen in each iteration. This implies that the requirement on the RIP constant becomes less restrictive when r increases. Such behavior seems to be natural but has not been reported for most of conventional methods. We also show that if r = 1, then by running more than K iterations, the performance guarantee of SSMP can be improved to δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[7.8K]</sub> ≤ 0.155. Furthermore, we show that under a suitable RIP condition, the reconstruction error of SSMP is upper bounded by a constant multiple of the noise power, which demonstrates the robustness of SSMP to measurement noise. Finally, from extensive numerical experiments, we show that SSMP outperforms conventional joint sparse recovery algorithms both in noiseless and noisy scenarios.