Toeplitz-Structured Compressed Sensing Matrices

The problem of recovering a sparse signal x Rn from a relatively small number of its observations of the form y = Ax Rk, where A is a known matrix and k « n, has recently received a lot of attention under the rubric of compressed sensing (CS) and has applications in many areas of signal processing such as data cmpression, image processing, dimensionality reduction, etc. Recent work has established that if A is a random matrix with entries drawn independently from certain probability distributions then exact recovery of x from these observations can be guaranteed with high probability. In this paper, we show that Toeplitz-structured matrices with entries drawn independently from the same distributions are also sufficient to recover x from y with high probability, and we compare the performance of such matrices with that of fully independent and identically distributed ones. The use of Toeplitz matrices in CS applications has several potential advantages: (i) they require the generation of only O(n) independent random variables; (ii) multiplication with Toeplitz matrices can be efficiently implemented using fast Fourier transform, resulting in faster acquisition and reconstruction algorithms; and (iii) Toeplitz-structured matrices arise naturally in certain application areas such as system identification.

1. INTRODUCTION

We begin by revisiting the problem of recovering a signal from linear observations of the form

$$y=Ax \quad: \quad \Vert x\Vert_{0}\leq m, \eqno{\hbox{(1)}}$$ where counts the number of non-zero entries in a vector, and is a known matrix. Of particular interest is the special case of highly underdetermined system, , that has applications in many areas of signal processing such as data compression, image processing, dimensionality reduction etc. and has recently received a lot of attention under the rubric of compressed sensing (CS) – starting in particular with some of the earlier works of Candes, Romberg and Tao [1], [2], [3] and Donoho [4].