Linear Inverse Problems with Norm and Sparsity Constraints

TL;DR

This paper introduces two novel algorithms, GAME and CLASH, for linear regression that jointly utilize convex and non-convex sparsity constraints, offering improved theoretical guarantees and empirical results.

Contribution

The paper presents innovative algorithms that combine convex and non-convex constraints for sparse recovery, advancing the state of the art.

Findings

Enhanced theoretical approximation guarantees

Superior empirical performance over existing methods

Effective joint use of convex and non-convex constraints

Abstract

We describe two nonconventional algorithms for linear regression, called GAME and CLASH. The salient characteristics of these approaches is that they exploit the convex $\ell_1$-ball and non-convex $\ell_0$-sparsity constraints jointly in sparse recovery. To establish the theoretical approximation guarantees of GAME and CLASH, we cover an interesting range of topics from game theory, convex and combinatorial optimization. We illustrate that these approaches lead to improved theoretical guarantees and empirical performance beyond convex and non-convex solvers alone.