Signal Recovery in Uncorrelated and Correlated Dictionaries Using Orthogonal Least Squares

TL;DR

This paper demonstrates that Orthogonal Least Squares (OLS) is a competitive and efficient method for sparse signal recovery, outperforming Orthogonal Matching Pursuit (OMP) especially in correlated dictionaries, with theoretical and numerical validation.

Contribution

The paper introduces an adaptation of OLS for compressed recovery, showing it can exactly recover sparse signals with fewer measurements and better performance in correlated dictionaries than existing greedy algorithms.

Findings

OLS achieves exact recovery with O(K log(N/K)) measurements.

OLS has similar computational complexity to OMP.

OLS outperforms OMP in correlated dictionary scenarios.

Abstract

Though the method of least squares has been used for a long time in solving signal processing problems, in the recent field of sparse recovery from compressed measurements, this method has not been given much attention. In this paper we show that a method in the least squares family, known in the literature as Orthogonal Least Squares (OLS), adapted for compressed recovery problems, has competitive recovery performance and computation complexity, that makes it a suitable alternative to popular greedy methods like Orthogonal Matching Pursuit (OMP). We show that with a slight modification, OLS can exactly recover a $K$-sparse signal, embedded in an $N$ dimensional space ($K<<N$) in $M=\mathcal{O}(K\log (N/K))$ no of measurements with Gaussian dictionaries. We also show that OLS can be easily implemented in such a way that it requires $\mathcal{O}(KMN)$ no of floating point operations similar to that of OMP. In this paper performance of OLS is also studied with sensing matrices with correlated dictionary, in which algorithms like OMP does not exhibit good recovery performance. We study the recovery performance of OLS in a specific dictionary called \emph{generalized hybrid dictionary}, which is shown to be a correlated dictionary, and show numerically that OLS has is far superior to OMP in these kind of dictionaries in terms of recovery performance. Finally we provide analytical justifications that corroborate the findings in the numerical illustrations.