Practical Compressive Sensing with Toeplitz and Circulant Matrices Wotao Yin a , Simon Morgan b , Junfeng Yang c , Yin Zhang a a Rice University, Dept. of Computational and Applied Mathematics, Houston, TX, USA. b New Mexico Consortium, Los Alamos, NM, USA. c Nanjing University, Department of Mathematics, Nanjing, Jiangsu, China. ABSTRACT Compressive sensing encodes a signal into a relatively small number of incoherent linear measurements. In theory, the optimal incoherence is achieved by completely random measurement matrices. However, such matrices are often difficult and costly to implement in hardware realizations. Random Toeplitz and circulant matrices can be easily (or even naturally) realized in various applications. This paper introduces fast algorithms for reconstructing signals from incomplete Toeplitz and circulant measurements. Computational results are presented to show that Toeplitz and circulant matrices are not only as effective as random matrices for signal encoding, but also permit much faster decoding. Keywords: Compressive Sensing, Circulant, Toeplitz, Signal Reconstruction, Random Matrix 1. INTRODUCTION Compressive sensing (CS), an emerging methodology brought to the research forefront by Donoho 1 and Candes, Romberg, and Tao, 2 acquires a compressible signal from a relatively small set of linear projections. Let ¯ x denote an n-dimensional real or complex vector that is sparse under a certain basis Ψ, and b := A¯ x represent a set of m linear projections of ¯ x. The basis pursuit problem BP: min x ‖Ψx‖ 1 s.t. Ax = b (1) has been known to return a sparse Ψx. Let k be the number of nonzeros in Ψ¯ x. It is shown 3, 4 that, when A is a Gaussian random or partial Fourier ensemble, BP returns a solution equal to ¯ x with high probability from m = O(k log(n/k)) or O(k log 4 (n)) linear measurements, respectively. In CS applications, the acquisition of the linear projections A¯ x requires a physical implementation. In most cases, the use of an i.i.d. Gaussian random matrix A is either impossible or overly expensive. This motivates the study of easily implementable CS matrices. Two such matrices are the Toeplitz and circulant matrices, which have been shown to be almost as effective as the Gaussian random matrix for CS encoding/decoding. Toeplitz and circulant matrices have the forms, respectively, T =      t n t n−1 ··· t 1 t n+1 t n ··· t 2 . . . . . . . . . t 2n−1 t 2n−2 ··· t n      and C =      t n t n−1 ··· t 1 t 1 t n ··· t 2 . . . . . . . . . t n−1 t n−2 ··· t n      , where every left-to-right descending diagonal is constant, i.e., T i,j = T i+1,j+1 . If T satisfies the additional property that t i = t n+i , ∀i, it is also a circulant matrix C . Since any (partial) Toeplitz matrix can be extended to a (partial) circulant matrix, our discussions below are based exclusively on circulant matrices. Using Toeplitz, rather than circulant, matrices will require some minor computation overhead. Send correspondence to the first author at wotao.yin@rice.edu.