Sparse recovery via Orthogonal Least-Squares under presence of Noise

TL;DR

This paper analyzes the Orthogonal Least-Squares (OLS) algorithm for recovering sparse signals from noisy measurements, establishing conditions for exact support recovery and demonstrating its efficiency with probabilistic guarantees.

Contribution

It provides new recovery guarantees for OLS under noise, improving existing bounds for OMP, and offers probabilistic analysis for random measurement matrices.

Findings

OLS recovers the true support in k iterations with high probability.

O( k log m ) measurements suffice for exact recovery.

The framework extends to Gaussian and Bernoulli measurement matrices.

Abstract

We consider the Orthogonal Least-Squares (OLS) algorithm for the recovery of a $m$-dimensional $k$-sparse signal from a low number of noisy linear measurements. The Exact Recovery Condition (ERC) in bounded noisy scenario is established for OLS under certain condition on nonzero elements of the signal. The new result also improves the existing guarantees for Orthogonal Matching Pursuit (OMP) algorithm. In addition, This framework is employed to provide probabilistic guarantees for the case that the coefficient matrix is drawn at random according to Gaussian or Bernoulli distribution where we exploit some concentration properties. It is shown that under certain conditions, OLS recovers the true support in $k$ iterations with high probability. This in turn demonstrates that ${\cal O}\left(k\log m\right)$ measurements is sufficient for exact recovery of sparse signals via OLS.