Corrupted Sensing: Novel Guarantees for Separating Structured Signals

TL;DR

This paper extends compressed sensing to corrupted measurements, providing geometric conditions and convex programming methods for exact and stable recovery of structured signals amidst corruption.

Contribution

It introduces new theoretical guarantees for signal recovery under structured corruption using Gaussian complexity and convex optimization, with practical bounds for various structures.

Findings

Theoretical conditions for exact recovery with structured corruption

Stable recovery guarantees with unstructured noise

Simulation results confirming phase transition predictions

Abstract

We study the problem of corrupted sensing, a generalization of compressed sensing in which one aims to recover a signal from a collection of corrupted or unreliable measurements. While an arbitrary signal cannot be recovered in the face of arbitrary corruption, tractable recovery is possible when both signal and corruption are suitably structured. We quantify the relationship between signal recovery and two geometric measures of structure, the Gaussian complexity of a tangent cone and the Gaussian distance to a subdifferential. We take a convex programming approach to disentangling signal and corruption, analyzing both penalized programs that trade off between signal and corruption complexity, and constrained programs that bound the complexity of signal or corruption when prior information is available. In each case, we provide conditions for exact signal recovery from structured corruption and stable signal recovery from structured corruption with added unstructured noise. Our simulations demonstrate close agreement between our theoretical recovery bounds and the sharp phase transitions observed in practice. In addition, we provide new interpretable bounds for the Gaussian complexity of sparse vectors, block-sparse vectors, and low-rank matrices, which lead to sharper guarantees of recovery when combined with our results and those in the literature.