On the Fundamental Recovery Limit of Orthogonal Least Squares

TL;DR

This paper establishes the exact recovery conditions for the orthogonal least squares (OLS) algorithm in sparse signal reconstruction, providing optimal RIP bounds and demonstrating their tightness.

Contribution

It derives the fundamental RIP-based recovery guarantees for OLS and proves their optimality, advancing theoretical understanding of sparse recovery limits.

Findings

OLS guarantees exact recovery under specific RIP conditions.

The RIP bounds for OLS are shown to be tight and optimal.

Counterexamples exist when RIP conditions are not met.

Abstract

Orthogonal least squares (OLS) is a classic algorithm for sparse recovery, function approximation, and subset selection. In this paper, we analyze the performance guarantee of the OLS algorithm. Specifically, we show that OLS guarantees the exact reconstruction of any $K$-sparse vector in $K$ iterations, provided that a sensing matrix has unit $\ell_{2}$-norm columns and satisfies the restricted isometry property (RIP) of order $K+1$ with \begin{align*} \delta_{K+1} &<C_{K} = \begin{cases} \frac{1}{\sqrt{K}}, & K=1, \\ \frac{1}{\sqrt{K+\frac{1}{4}}}, & K=2, \\ \frac{1}{\sqrt{K+\frac{1}{16}}}, & K=3, \\ \frac{1}{\sqrt{K}}, & K \ge 4. \end{cases} \end{align*} Furthermore, we show that the proposed guarantee is optimal in the sense that if $\delta_{K+1} \ge C_{K}$, then there exists a counterexample for which OLS fails the recovery.