Tight Sufficient Conditions on Exact Sparsity Pattern Recovery

TL;DR

This paper derives tight, sufficient information-theoretic conditions for exact support recovery of sparse signals from noisy Gaussian measurements, applicable to both exactly and approximately sparse vectors.

Contribution

It provides a new tight, sufficient condition for support recovery in noisy Gaussian measurement systems, extending to approximately sparse signals.

Findings

Derived a tight, sufficient condition for exact support recovery.

Compared new bounds with existing recovery conditions.

Extended results to approximately sparse signals.

Abstract

A noisy underdetermined system of linear equations is considered in which a sparse vector (a vector with a few nonzero elements) is subject to measurement. The measurement matrix elements are drawn from a Gaussian distribution. We study the information-theoretic constraints on exact support recovery of a sparse vector from the measurement vector and matrix. We compute a tight, sufficient condition that is applied to ergodic wide-sense stationary sparse vectors. We compare our results with the existing bounds and recovery conditions. Finally, we extend our results to approximately sparse signals.