The springback penalty for robust signal recovery

TL;DR

The paper introduces the springback penalty, a weakly convex function, for robust signal recovery from incomplete and noisy measurements, offering theoretical guarantees and computational advantages over existing methods.

Contribution

It proposes the springback penalty, establishes its exact and stable recovery properties, and develops an efficient difference-of-convex algorithm for practical implementation.

Findings

Sharper recovery bounds in high-noise or limited-measurement scenarios

Numerical robustness across various sensing matrix coherences

Effective recovery with severely incomplete and inaccurate measurements

Abstract

We propose a new penalty, the springback penalty, for constructing models to recover an unknown signal from incomplete and inaccurate measurements. Mathematically, the springback penalty is a weakly convex function. It bears various theoretical and computational advantages of both the benchmark convex $\ell_1$ penalty and many of its non-convex surrogates that have been well studied in the literature. We establish the exact and stable recovery theory for the recovery model using the springback penalty for both sparse and nearly sparse signals, respectively, and derive an easily implementable difference-of-convex algorithm. In particular, we show its theoretical superiority to some existing models with a sharper recovery bound for some scenarios where the level of measurement noise is large or the amount of measurements is limited. We also demonstrate its numerical robustness regardless of the varying coherence of the sensing matrix. The springback penalty is particularly favorable for the scenario where the incomplete and inaccurate measurements are collected by coherence-hidden or -static sensing hardware due to its theoretical guarantee of recovery with severe measurements, computational tractability, and numerical robustness for ill-conditioned sensing matrices.