Compressive covariance sampling

Abstract

Abstract-Most research efforts in the field of compressed sensing have been pointed towards analyzing sampling and reconstruction techniques for sparse signals, where sampling rates below the Nyquist rate can be reached. When only second-order statistics or, equivalently, covariance information is of interest, perfect signal reconstruction is not required and rate reductions can be achieved even for non-sparse signals. This is what we will refer to as compressive covariance sampling. In this paper, we will study minimum-rate compressive covariance sampling designs within the class of non-uniform samplers. Necessary and sufficient conditions for perfect covariance reconstruction will be provided and connections to the well-known sparse ruler problem will be highlighted. I. PROBLEM STATEMENTConsider the problem of estimating the covariance matrix Σ of a random vector x with components x[n] when it is known that Σ is a linear combination of the matrices in the set S. This problem has a long history and a wide range of applications [1], [2]. Now consider a modification of the same problem where only a subset of the samples x[n] is available, i.e., when the observations are y[m] = x[n m ] for some I = {n 0 , n 1 , . . .} ⊂ Z. Intuitively, if the grid defined by I is dense enough, then this estimation problem, which we label as compressive covariance sampling, can still be solved.More specifically, the above sample selection produces a transformation of the problem: the vector y collecting the compressed observations y[m], has a covariance matrixΣ, which is a linear combination of the matrices inS. The coefficients in the linear combination are in both problems the same so that solving the latter amounts to solving the former. In this paper we address the optimal design of the index set I so that the coefficients can be estimated, i.e., we look for sets of indices with a minimum number of elements guaranteeing that these parameters remain statistically identifiable. A. Relation to Compressive SamplingRecent interest in sampling signals below their Nyquist rate owes to compressive sampling (or compressed sensing) [3], which has motivated a great deal of research efforts in the last few years. In compressive sampling we are interested in reducing the number of samples by focusing on a special family of signals referred to as sparse. These signals are intrinsically redundant, so that this reduction does not entail any information loss if properly done.Non-uniform sampling [4] is a particular case of compressive sampling. Mathematically, this process can be described as first acquiring a continuous-index signal x(t) at rate 1/T , resulting in the sequence x[n] = x(nT ), and then downsampling x[n] to obtain y[m] according to a set of indices I, in a periodic or non-periodic fashion (periodic non-uniform sampling is also known as multi-coset sampling).Either in the context of compressive sampling (the nonuniform sampling version) or in the context of what we call compressive covariance sampling, the design of the set I has ...

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“…Proof: See Appendix A. Remarks: We note that the recovery condition (28) is slightly relaxed than the sparse ruler condition given in [18], and is equivalent to the recovery condition given in [26] which is termed as the circular sparse ruler. For the sparse ruler condition [18], it requires the set {0, 1, .…”

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“…Compared with traditional Nyquist sampling theory, the number of samples required in CS can be made much smaller by exploiting the signal sparsity property exhibited in a certain domain. Periodic Nonuniform Sampling (PNS) is a popular approach among those techniques, specially when only covariance information is of interest [34]. In this paper we consider the sub-Nyquist PNS strategy known as multicoset (MC) sampling [35], [36].…”

Generalized nested sampling for compression and exact recovery of symmetric Toeplitz matrices

“…In this paper, we assume such a special structure of the covariance matrix by constraining it to have a Toeplitz structure. A Toeplitz structured covariance matrix describes the covariance matrix of Wide Sense Stationary random processes and is very common in many signal processing applications requiring spectral estimation [4], [5].…”

“…In [5], the Toeplitz form of covariance matrix is considered and the sample size is also proved to be of the order of O( √ N ). In both [6] and [5], sparse ruler is used to construct the sampling matrix, which does not have a closed form solution for any given integer N . In [1], a high dimensional covariance matrix Σ ∈ R N,N is sketched using quadratic measurements of the form Y = WΣW with measurement matrix W andW ∈ R S,N .…”

“…Σ is assumed to have a particular property named distributed sparsity such that every row and every column has only a few non-zeros entries, and the sample size is proved to be O( [2], a similar problem of compressive covariance estimation is considered and for general Toeplitz matrix, a sample size of O( √ N polylogN ) is shown to be sufficient for successful recovery. A common feature of all existing work on compressive covariance sketching and estimation is that they do not indicate how much additional compression is possible (beyond a compression factor of O( √ N ) when the Toeplitz matrix is sparse [1], [5], [2]. This brings us to the main contributions of our paper.…”