  Orthogonal least squares (OLS)-type algorithms are efficient in
reconstructing sparse signals, which include the well-known OLS, multiple OLS
(MOLS) and block OLS (BOLS). In this paper, we first investigate the noiseless
exact recovery conditions of these algorithms. Specifically, based on mutual
incoherence property (MIP), we provide theoretical analysis of OLS and MOLS to
ensure that the correct nonzero support can be selected during the iterative
procedure. Nevertheless, theoretical analysis for BOLS utilizes the block-MIP
to deal with the block sparsity. Furthermore, the noiseless MIP-based analyses
are extended to the noisy scenario. Our results indicate that for K-sparse
signals, when MIP or SNR satisfies certain conditions, OLS and MOLS obtain
reliable reconstruction in at most K iterations, while BOLS succeeds in at most
(K/d) iterations where d is the block length. It is shown that our derived
theoretical results improve the existing ones, which are verified by simulation
tests.
