Recovery of sparse signals via Branch and Bound Least-Squares

We present an algorithm, referred to as Branch and Bound Least-Squares (BBLS), for the recovery of sparse signals from a few linear combinations of their entries. Sparse signal reconstruction is readily cast as the problem of finding a sparse solution to an underdetermined system of linear equations. To solve it, BBLS employs an efficient search strategy of traversing a tree whose nodes represent the columns of the coefficient matrix and selects a subset of those columns by relying on Orthogonal Least-Squares (OLS) procedure. We state sufficient conditions under which in noise-free settings BBLS with high probability constructs a tree path which corresponds to the true support of the unknown sparse signal. Moreover, we empirically demonstrate that BBLS provides performance superior to that of existing algorithms in terms of accuracy, running time, or both. In the scenarios where the columns of the coefficient matrix are characterized by high correlation, BBLS is particularly beneficial and significantly outperforms existing methods.