DETERMINISTIC COMPRESSED-SENSING MATRICES: WHERE TOEPLITZ MEETS GOLAY Kezhi Li , Cong Ling , and Lu Gan†
Imperial College London, UK; † Brunel University, UK {k.li08, c.ling}@imperial.ac.uk;
[email protected]
ABSTRACT Recently, the statistical restricted isometry property (STRIP) has been formulated to analyze the performance of deterministic sampling matrices for compressed sensing. In this paper, a class of deterministic matrices which satisfy STRIP with overwhelming probability are proposed, by taking advantage of concentration inequalities using Stein’s method. These matrices, called orthogonal symmetric Toeplitz matrices (OSTM), guarantee successful recovery of all but an exponentially small fraction of K-sparse signals. Such matrices are deterministic, Toeplitz, and easy to generate. We derive the STRIP performance bound by exploiting the specific properties of OSTM, and obtain the near-optimal bound by setting the underlying sign sequence of OSTM as the Golay sequence. Simulation results show that these deterministic sensing matrices can offer reconstruction performance similar to that of random matrices. Index Terms— compressed sensing, Golay sequence, statistical restricted isometry property, Toeplitz matrix. I. INTRODUCTION Compressed Sensing (CS) is a new framework for simultaneous sampling and compression of signals that utilizes sparsity in representations to reduce the number of linear measurements needed for signal encoding. CS mainly involves a problem of recovering a K-sparse signal x ∈ RN from a relatively small number of its measurements in the form (1) y = Φx ∈ RM where Φ is the sensing matrix and M N . It was established in [1], [2] that for a matrix Φ to be a CS sensing matrix, it is sufficient that it satisfies the restricted isometry property (RIP). The RIP makes sensing matrix act as a near isometry on all K-sparse vectors. An M × N measurement matrix Φ has the RIP with parameters (K, δ) for δ ∈ (0, 1) means it satisfies (1 − δ)x2 ≤ Φx2 ≤ (1 + δ)x2 , for all x ∈ Ω, (2) where Ω denotes the set of all length-N vectors with K non-zero coefficients.
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It has been established by Donoho and by Candes, Romberg and Tao [1]–[3] that random matrices whose entries are drawn independently from certain probability distributions satisfy RIP with overwhelming probability. However, such random matrices are impractical for large N , since they require high computational and storage complexity. Statistical restricted isometry property (STRIP) was recently formulated by several authors, particulary by Calderbank et al. [4] to analyze the performance of deterministic sampling matrices for compressed sensing. Compared with random ones, deterministic matrices obtain many advantages in complexity, reconstruction efficiency and storage. When a deterministic sensing matrix satisfies STRIP, it means that (2) holds with respect to a uniform distribution of the vectors x among all K-sparse vectors in RN of the same norm. It is a weaker version of RIP actually, and indicates the expected-case performance only. Yet it could be proved that when certain deterministic sensing matrices satisfy STRIP with overwhelming probability, the original signal could be uniquely recovered in most cases. In this paper, a class of deterministic matrices which satisfy STRIP with overwhelming probability are proposed, by taking advantage of concentration inequalities using Stein’s method [5]. Because there is no practical algorithm to verify RIP, deterministic sensing matrices with a guaranteed theoretical bound is considered as a feasible method for compressed sensing. These matrices, termed orthogonal symmetric Toeplitz matrices (OSTM) here, guarantee successful recovery of all but an exponentially small fraction of K-sparse signals. We derive the STRIP performance bound by exploiting the specific properties of OSTM, and obtain the near-optimal reconstruction performance by using Golay sequences [6] as the underlying sign sequences to generate OSTM. Sampling with OSTM can be implemented using the fast Fourier transform (FFT) efficiently, and its Toeplitz structure arises naturally in many application areas. II. PROPERTIES OF OSTM An M × N sensing matrix based on OSTM can be constructed like this: 1) Use a binary sequence s of length N/2 to form the sign
ICASSP 2011
sequence σ = [s1 , · · · , sN/2 , ±s1 , sN/2 , · · · , s2 ],
(3)
and apply inverse FFT (IFFT) to the sign sequence to obtain g with length N [7]: g = IFFT(σ).
(4)
2) Let the elements of g be the first row of OSTM, and follow the circulant property to construct the N × N matrix ΦN . 3) Choose M rows and normalize it by multiplying N/M to form the M × N sensing matrix Φ. After the second step, it can be proved that the N × N matrix ΦN is orthogonal and Toeplitz. The sign of s1 in the middle of σ depends on the parity of the location of ϕ1 , which is the largest value in g. The specific structure of the sign sequence is a requirement of OSTM [7]. Since there are 2N/2 binary sequences s of length N/2, clearly some are better than others. In [8], the bound based on Chebyshev’s inequality was too loose to capture the impact of s. In this paper, we will significantly improve the bound by exploiting Stein’s method [5]. This will enable us to achieve near-optimal performance by using a Golay’s complementary sequence as s of the sign sequence. Several properties of the OSTM are also very important in the proof: 1) All the elements of a row or a column in the matrix are constituted by the same sequence; 2) The rows and columns are all orthogonal, and all row-sums are 1 before normalization. Compared to Romberg’s random convolution [9] and Calderbank et al.’s deterministic matrices [4], the proposed OSTM has such advantages: 1) all elements are real and deterministic; 2) sensing matrices are easier to generate; 3) the Toeplitz structure takes less storage space and less calculation; 4) the Toeplitz structure permits an FFT-based implementation for fast recovery algorithms, and the matrices are well suited to some applications that are inherently Toeplitz, such as channel estimation and Terahertz imaging. III. MAIN THEOREM AND THE PROOF The main theorem is presented as follow: Theorem 1: Let x be a N -length, K-sparse signal with non-zero coefficientsx1 , x2 , x3 , · · · xK . Assume that x K has zero-mean (i.e. i=1 xi = 0) and the positions of the K non-zero entries are equiprobable. Let Φ be an M × N deterministic normalized sensing matrix by selecting M rows arbitrarily from a N × N OSTM times a normalized N . Then parameter M (5) E Φx2 = x2 , and the following inequality holds: P Φx2 − x2 < δx2 M δ2 ≥ 1 − 2 exp − 8C1 · K
(6)
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M when K ≤ C0 (δ) log . Here C0 (δ) is a constant depending MN 2 on δ only. C1 = ζ=1 ϕζ , which is the sum of the M largest values ϕ1 , ϕ2 , . . . , ϕM in g. The Golay sequence is used in this paper to obtain a small value of C1 . The binary Golay sequences are known to exist for all lengths 2α1 10α2 26α3 , α1 , α2 , α3 are nonnegative numbers [6]. The reason we choose a Golay sequence as sign sequence is it has a special autocorrelation property, such that the envelope of its Fourier transform is limited to a small range relative to the mean. Formally, the peak-tomean envelop power ratio (PMEPR) of a Golay sequence s = (s1 , . . . , sN/2 ) is less than 2 [10]:
PMEPRs = max N/2
0≤ω
