Orthogonal symmetric Toeplitz matrices for compressed sensing: Statistical isometry property

TL;DR

This paper introduces orthogonal symmetric Toeplitz matrices (OSTM) for compressed sensing, demonstrating their ability to satisfy statistical RIP and achieve reconstruction performance comparable to random matrices, using Golay sequences.

Contribution

It proposes a novel deterministic matrix construction for compressed sensing based on OSTM and analyzes their statistical RIP using Stein's method, highlighting their efficiency and effectiveness.

Findings

OSTM can satisfy statistical RIP for most signals with given sparsity.

Golay sequences enable OSTM to meet RIP conditions.

Simulation shows OSTM's reconstruction performance rivals random matrices.

Abstract

Recently, the statistical restricted isometry property (RIP) has been formulated to analyze the performance of deterministic sampling matrices for compressed sensing. In this paper, we propose the usage of orthogonal symmetric Toeplitz matrices (OSTM) for compressed sensing and study their statistical RIP by taking advantage of Stein's method. In particular, we derive the statistical RIP performance bound in terms of the largest value of the sampling matrix and the sparsity level of the input signal. Based on such connections, we show that OSTM can satisfy the statistical RIP for an overwhelming majority of signals with given sparsity level, if a Golay sequence used to generate the OSTM. Such sensing matrices are deterministic, Toeplitz, and efficient to implement. Simulation results show that OSTM can offer reconstruction performance similar to that of random matrices.