  We consider the greedy algorithms for the joint recovery of high-dimensional
sparse signals based on the block multiple measurement vector (BMMV) model in
compressed sensing (CS). To this end, we first put forth two versions of
simultaneous block orthogonal least squares (S-BOLS) as the baseline for the
OLS framework. Their cornerstone is to sequentially check and select the
support block to minimize the residual power. Then, parallel performance
analysis for the existing simultaneous block orthogonal matching pursuit
(S-BOMP) and the two proposed S-BOLS algorithms is developed. It indicates that
under the conditions based on the mutual incoherence property (MIP) and the
decaying magnitude structure of the nonzero blocks of the signal, the
algorithms select all the significant blocks before possibly choosing incorrect
ones. In addition, we further consider the problem of sufficient data volume
for reliable recovery, and provide its MIP-based bounds in closed-form. These
results together highlight the key role of the block characteristic in
addressing the weak-sparse issue, i.e., the scenario where the overall sparsity
is too large. The derived theoretical results are also universally valid for
conventional block-greedy algorithms and non-block algorithms by setting the
number of measurement vectors and the block length to 1, respectively.
