A Unified Recovery of Structured Signals Using Atomic Norm

TL;DR

This paper presents a unified convex optimization framework for recovering structured signals, such as sparse vectors and low-rank matrices, from limited linear measurements, with guarantees based on null space properties.

Contribution

It introduces a general null space property condition ensuring stable recovery and demonstrates its high-probability satisfaction with subgaussian measurements, extending to sparse frames and low-rank matrices.

Findings

Convex programming can stably recover signals with certain null space properties.

Null space property is satisfied with high probability for subgaussian measurements.

New results for sparse signals in frames and low-rank matrix recovery.

Abstract

In many applications we seek to recover signals from linear measurements far fewer than the ambient dimension, given the signals have exploitable structures such as sparse vectors or low rank matrices. In this paper we work in a general setting where signals are approximately sparse in an so called atomic set. We provide general recovery results stating that a convex programming can stably and robustly recover signals if the null space of the sensing map satisfies certain properties. Moreover, we argue that such null space property can be satisfied with high probability if each measurement is subgaussian even when the number of measurements are very few. Some new results for recovering signals sparse in a frame, and recovering low rank matrices are also derived as a result.