Revisiting matching pursuit: Beyond approximate submodularity

We study the problem of selecting a subset of vectors from a large set to obtain the best signal representation over a family of functions. Although greedy methods have been widely used to tackle this problem and many of those have been analyzed under the lens of (weak) submodularity, none of these algorithms are explicitly devised using such a functional property. Here, we revisit the vector-selection problem and introduce a function that is shown to be submodular in expectation. This function not only guarantees near-optimality through a greedy algorithm in expectation but also alleviates the existing deficiencies in commonly used matching pursuit (MP) algorithms. We further show the relation between the single-point-estimate version of the proposed greedy algorithm and MP variants. Moreover, we discuss extending the signal representation problem to instances with knapsack and matroid constraints. Our theoretical findings are supported by numerical experiments on the angle of arrival estimation problem, a typical signal representation task, demonstrating the benefits of our method compared to traditional MP algorithms.