Recovery Conditions of Sparse Signals Using Orthogonal Least Squares-Type Algorithms

TL;DR

This paper analyzes the exact recovery conditions of OLS-type algorithms for sparse signals, providing theoretical guarantees based on mutual incoherence properties, and extends these results to noisy scenarios with improved bounds verified by simulations.

Contribution

It offers new theoretical analysis of OLS, MOLS, and BOLS algorithms using mutual incoherence properties, including noisy case extensions and improved recovery guarantees.

Findings

OLS and MOLS reliably recover sparse signals within K iterations.

BOLS recovers block sparse signals in at most (K/d) iterations.

Theoretical results are validated through simulations showing improved bounds.

Abstract

Orthogonal least squares (OLS)-type algorithms are efficient in reconstructing sparse signals, which include the well-known OLS, multiple OLS (MOLS) and block OLS (BOLS). In this paper, we first investigate the noiseless exact recovery conditions of these algorithms. Specifically, based on mutual incoherence property (MIP), we provide theoretical analysis of OLS and MOLS to ensure that the correct nonzero support can be selected during the iterative procedure. Nevertheless, theoretical analysis for BOLS utilizes the block-MIP to deal with the block sparsity. Furthermore, the noiseless MIP-based analyses are extended to the noisy scenario. Our results indicate that for K-sparse signals, when MIP or SNR satisfies certain conditions, OLS and MOLS obtain reliable reconstruction in at most K iterations, while BOLS succeeds in at most (K/d) iterations where d is the block length. It is shown that our derived theoretical results improve the existing ones, which are verified by simulation tests.