Probabilistic Recovery Guarantees for Sparsely Corrupted Signals

TL;DR

This paper introduces probabilistic guarantees for recovering sparse signals corrupted by sparse interference, applicable in practical scenarios, even when sparsity levels are high relative to measurements.

Contribution

It provides novel probabilistic recovery guarantees for sparse signals with sparse interference, considering varying prior knowledge and coherence properties of dictionaries.

Findings

High-probability recovery even with linear sparsity scaling

Guarantees depend on dictionary coherence parameters

Applicable to practical signal processing scenarios

Abstract

We consider the recovery of sparse signals subject to sparse interference, as introduced in Studer et al., IEEE Trans. IT, 2012. We present novel probabilistic recovery guarantees for this framework, covering varying degrees of knowledge of the signal and interference support, which are relevant for a large number of practical applications. Our results assume that the sparsifying dictionaries are characterized by coherence parameters and we require randomness only in the signal and/or interference. The obtained recovery guarantees show that one can recover sparsely corrupted signals with overwhelming probability, even if the sparsity of both the signal and interference scale (near) linearly with the number of measurements.