Sparsity Pattern Recovery in Bernoulli-Gaussian Signal Model

TL;DR

This paper investigates the problem of recovering the support of sparse signals modeled with Bernoulli-Gaussian distributions in compressive sensing, providing bounds and conditions for accurate support detection.

Contribution

It introduces probabilistic bounds and sufficient conditions for perfect support recovery in Bernoulli-Gaussian models, advancing theoretical understanding.

Findings

Bound on energy in missed support related to noise projection

Sufficient conditions for no misdetection and false alarms

Support recovery performance characterized under probabilistic model

Abstract

In compressive sensing, sparse signals are recovered from underdetermined noisy linear observations. One of the interesting problems which attracted a lot of attention in recent times is the support recovery or sparsity pattern recovery problem. The aim is to identify the non-zero elements in the original sparse signal. In this article we consider the sparsity pattern recovery problem under a probabilistic signal model where the sparse support follows a Bernoulli distribution and the signal restricted to this support follows a Gaussian distribution. We show that the energy in the original signal restricted to the missed support of the MAP estimate is bounded above and this bound is of the order of energy in the projection of the noise signal to the subspace spanned by the active coefficients. We also derive sufficient conditions for no misdetection and no false alarm in support recovery.