Exact Support and Vector Recovery of Constrained Sparse Vectors via Constrained Matching Pursuit

Matching pursuit, especially its orthogonal version (OMP) and variations, is a greedy algorithm widely used in signal processing, compressed sensing, and sparse modeling. Inspired by constrained sparse signal recovery, this paper proposes a constrained matching pursuit algorithm and develops conditions for exact support and vector recovery on constraint sets via this algorithm. We show that exact recovery via constrained matching pursuit not only depends on a measurement matrix but also critically relies on a constraint set. We thus identify an important class of constraint sets, called coordinate projection admissible set, or simply CP admissible sets; analytic and geometric properties of these sets are established. We study exact vector recovery on convex, CP admissible cones for a fixed support. We provide sufficient exact recovery conditions for a general support as well as necessary and sufficient recovery conditions when a support has small size. As a byproduct, we construct a nontrivial counterexample to a renowned necessary condition of exact recovery via the OMP for a support of size three. Moreover, using the properties of convex CP admissible sets and convex optimization techniques, we establish sufficient conditions for uniform exact recovery on convex CP admissible sets in terms of the restricted isometry-like constant and the restricted orthogonality-like constant.