  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.
