Nearly Optimal Bounds for Orthogonal Least Squares

TL;DR

This paper establishes nearly optimal theoretical bounds for the orthogonal least squares algorithm, showing when it can and cannot exactly recover sparse signals based on the restricted isometry property of the sampling matrix.

Contribution

The paper provides the first sharp bounds on the RIP constants under which OLS guarantees exact support recovery of sparse signals.

Findings

OLS recovers support exactly if elta_{K+1} < 1/(K+1)

OLS may fail to recover support if elta_{K+1}  rac{1}{(K+1/4)}

Bounds are nearly optimal for the RIP-based recovery guarantees.

Abstract

In this paper, we study the orthogonal least squares (OLS) algorithm for sparse recovery. On the one hand, we show that if the sampling matrix $\mathbf{A}$ satisfies the restricted isometry property (RIP) of order $K + 1$ with isometry constant $$ \delta_{K + 1} < \frac{1}{\sqrt{K+1}}, $$ then OLS exactly recovers the support of any $K$-sparse vector $\mathbf{x}$ from its samples $\mathbf{y} = \mathbf{A} \mathbf{x}$ in $K$ iterations. On the other hand, we show that OLS may not be able to recover the support of a $K$-sparse vector $\mathbf{x}$ in $K$ iterations for some $K$ if $$ \delta_{K + 1} \geq \frac{1}{\sqrt{K+\frac{1}{4}}}. $$