The Convergence Guarantees of a Non-convex Approach for Sparse Recovery

TL;DR

This paper introduces a non-convex sparse recovery method with convergence guarantees, demonstrating that under certain conditions, it reliably recovers sparse signals with errors proportional to noise and step size.

Contribution

It provides the first convergence guarantees for a non-convex sparse recovery algorithm using weak convexity and a projected subgradient method.

Findings

Recovery error is linear in noise and step size.

Algorithm converges under bounded non-convexity.

Numerical results confirm theoretical predictions.

Abstract

In the area of sparse recovery, numerous researches hint that non-convex penalties might induce better sparsity than convex ones, but up until now those corresponding non-convex algorithms lack convergence guarantees from the initial solution to the global optimum. This paper aims to provide performance guarantees of a non-convex approach for sparse recovery. Specifically, the concept of weak convexity is incorporated into a class of sparsity-inducing penalties to characterize the non-convexity. Borrowing the idea of the projected subgradient method, an algorithm is proposed to solve the non-convex optimization problem. In addition, a uniform approximate projection is adopted in the projection step to make this algorithm computationally tractable for large scale problems. The convergence analysis is provided in the noisy scenario. It is shown that if the non-convexity of the penalty is below a threshold (which is in inverse proportion to the distance between the initial solution and the sparse signal), the recovered solution has recovery error linear in both the step size and the noise term. Numerical simulations are implemented to test the performance of the proposed approach and verify the theoretical analysis.