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Matching pursuit, especially its orthogonal version (OMP) and variations, isa greedy algorithm widely used in signal processing, compressed sensing, andsparse modeling. Inspired by constrained sparse signal recovery, this paperproposes a constrained matching pursuit algorithm and develops conditions forexact support and vector recovery on constraint sets via this algorithm. Weshow that exact recovery via constrained matching pursuit not only depends on ameasurement matrix but also critically relies on a constraint set. We thusidentify an important class of constraint sets, called coordinate projectionadmissible set, or simply CP admissible sets; analytic and geometric propertiesof these sets are established. We study exact vector recovery on convex, CPadmissible cones for a fixed support. We provide sufficient exact recoveryconditions for a general support as well as necessary and sufficient recoveryconditions when a support has small size. As a byproduct, we construct anontrivial counterexample to a renowned necessary condition of exact recoveryvia the OMP for a support of size three. Moreover, using the properties ofconvex CP admissible sets and convex optimization techniques, we establishsufficient conditions for uniform exact recovery on convex CP admissible setsin terms of the restricted isometry-like constant and the restrictedorthogonality-like constant.

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