Sharp Sufficient Conditions on Exact Sparsity Pattern Recovery

TL;DR

This paper establishes sharp, non-asymptotic conditions under which the exact sparsity pattern of a signal can be reliably recovered from noisy linear measurements, matching known necessary conditions.

Contribution

It provides non-asymptotic upper bounds on the probability of incorrect sparsity pattern recovery and asymptotically sharp sufficient conditions for exact recovery with Gaussian matrices.

Findings

Derived non-asymptotic bounds for decoding error probability.

Established asymptotically sharp sufficient conditions for exact recovery.

Conditions align with previously known necessary conditions.

Abstract

Consider the $n$-dimensional vector $y=X\be+\e$, where $\be \in \R^p$ has only $k$ nonzero entries and $\e \in \R^n$ is a Gaussian noise. This can be viewed as a linear system with sparsity constraints, corrupted by noise. We find a non-asymptotic upper bound on the probability that the optimal decoder for $\beta$ declares a wrong sparsity pattern, given any generic perturbation matrix $X$. In the case when $X$ is randomly drawn from a Gaussian ensemble, we obtain asymptotically sharp sufficient conditions for exact recovery, which agree with the known necessary conditions previously established.