Compressive Principal Component Pursuit

TL;DR

This paper addresses the challenge of recovering a matrix composed of low-rank and sparse components from limited linear measurements, with theoretical guarantees for exact recovery under random measurement conditions.

Contribution

It provides a rigorous analysis of a convex heuristic for decomposing superimposed low-rank and sparse signals in compressed sensing, with conditions for exact recovery.

Findings

Exact recovery when measurements exceed degrees of freedom by a polylogarithmic factor

Analysis introduces new ideas relevant to compressed sensing of structured signals

Applicable to high-dimensional signals like videos and hyperspectral images

Abstract

We consider the problem of recovering a target matrix that is a superposition of low-rank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured high-dimensional signals such as videos and hyperspectral images, as well as in the analysis of transformation invariant low-rank recovery. We analyze the performance of the natural convex heuristic for solving this problem, under the assumption that measurements are chosen uniformly at random. We prove that this heuristic exactly recovers low-rank and sparse terms, provided the number of observations exceeds the number of intrinsic degrees of freedom of the component signals by a polylogarithmic factor. Our analysis introduces several ideas that may be of independent interest for the more general problem of compressed sensing and decomposing superpositions of multiple structured signals.