  In this paper, we address the sparse multiple measurement vector (MMV)
problem where the objective is to recover a set of sparse nonzero row vectors
or indices of a signal matrix from incomplete measurements. Ideally, regardless
of the number of columns in the signal matrix, the sparsity (k) plus one
measurements is sufficient for the uniform recovery of signal vectors for
almost all signals, i.e., excluding a set of Lebesgue measure zero. To approach
the "k+1" lower bound with computational efficiency even when the rank of
signal matrix is smaller than k, we propose a greedy algorithm called Two-stage
orthogonal Subspace Matching Pursuit (TSMP) whose theoretical results approach
the lower bound with less restriction than the Orthogonal Subspace Matching
Pursuit (OSMP) and Subspace-Augmented MUltiple SIgnal Classification (SA-MUSIC)
algorithms. We provide non-asymptotical performance guarantees of OSMP and TSMP
by covering both noiseless and noisy cases. Variants of restricted isometry
property and mutual coherence are used to improve the performance guarantees.
Numerical simulations demonstrate that the proposed scheme has low complexity
and outperforms most existing greedy methods. This shows that the minimum
number of measurements for the success of TSMP converges more rapidly to the
lower bound than the existing methods as the number of columns of the signal
matrix increases.
