Recursive Sparse Recovery in Large but Structured Noise - Part 2

TL;DR

This paper introduces ReProCS-cPCA, a recursive method for accurately recovering sparse signals from measurements corrupted by structured, dense noise that lies in a low-dimensional subspace, with high probability guarantees.

Contribution

The paper proposes a novel recursive algorithm, ReProCS-cPCA, capable of exact support recovery of sparse signals amidst structured noise with minimal assumptions.

Findings

Exact support recovery with high probability

Bounded reconstruction errors for signals and noise

Effective handling of slowly changing noise subspace

Abstract

We study the problem of recursively recovering a time sequence of sparse vectors, St, from measurements Mt := St + Lt that are corrupted by structured noise Lt which is dense and can have large magnitude. The structure that we require is that Lt should lie in a low dimensional subspace that is either fixed or changes "slowly enough"; and the eigenvalues of its covariance matrix are "clustered". We do not assume any model on the sequence of sparse vectors. Their support sets and their nonzero element values may be either independent or correlated over time (usually in many applications they are correlated). The only thing required is that there be some support change every so often. We introduce a novel solution approach called Recursive Projected Compressive Sensing with cluster-PCA (ReProCS-cPCA) that addresses some of the limitations of earlier work. Under mild assumptions, we show that, with high probability, ReProCS-cPCA can exactly recover the support set of St at all times; and the reconstruction errors of both St and Lt are upper bounded by a time-invariant and small value.