Sparsity order estimation for single snapshot compressed sensing

Abstract

In this paper we discuss the estimation of the spar-sity order for a Compressed Sensing scenario where only a single snapshot is available. We demonstrate that a specific design of the sensing matrix based on Khatri-Rao products enables us to transform this problem into the estimation of a matrix rank in the presence of additive noise. Thereby, we can apply existing model order selection algorithms to determine the sparsity order. The matrix is a rearranged version of the observation vector which can be constructed by concatenating a series of non-overlapping or overlapping blocks of the original observation vector. In both cases, a Khatri-Rao structured measurement matrix is required with the main difference that in the latter case, one of the factors must be a Vandermonde matrix. We discuss the choice of the parameters and show that an increasing amount of block overlap improves the sparsity order estimation but it increases the coherence of the sensing matrix. We also explain briefly that the proposed measurement matrix design introduces certain multilinear structures into the observations which enables us to apply tensor-based signal processing, e.g., for enhanced denoising or improved sparsity order estimation. © 2014 IEEE

Tensor-based sparsity order estimation for big data applications

“…As explained in [6], for the purpose of sparsity order estimation, we can segment each single observation vector y(t) into B overlapping smaller vectors of size m, and obtain an observation matrix We can then show that for a K-sparse signal we have rank{Y (t)} = K if and only if we choose Ψ = (C ◇ Φ 0 ) ⋅ A H where ◇ represents the KhatriRao product (columnwise Kronecker product) of two matrices, and Φ 0 ∈ C p×N and C ∈ C M p ×N both have a Kruskal-rank 1 of at least K. In addition, if p < m, which means that adjacent blocks are overlapping, C needs to be a Vandermonde matrix.…”

“…As shown in [6] this is possible if and only if the sensing matrix is equal to the Khatri-Rao product of two full rank matrices. Moreover, when the columns of the observation matrix overlap, one of the two factors needs to be a Vandermonde matrix.…”

“…The sparsity structure in data is a crucial information to allow a drastic reduction on data storage and communication in big data applications. The authors in [6] have recently shown that if the data possesses a sparse representation in a suitable domain, one can estimate the sparsity order directly from the compressed data obtained by using a Khatri-Rao structured measurement matrix for compressed sensing. The structure of the measurement matrix allows to rearrange the multiplesnapshot observation vectors into a fourth-order tensor.…”

“…In [6], we have shown that the sparsity order estimation (SOE) problem can be transferred to the problem of estimating the rank of an observation matrix constructed from rearranging the low-dimensional observation vector into a matrix form. As shown in [6] this is possible if and only if the sensing matrix is equal to the Khatri-Rao product of two full rank matrices.…”

Simultaneous Signal Subspace Rank and Model Selection with an Application to Single-snapshot Source Localization

“…So far most research has focused on SSR task and much less attention has been paid to solve the detection task. Some existing detection approaches in the literature can be found e.g., in [3], [4]. Typically, most SSR (e.g., greedy pursuit methods [5]) presume that k * is known or an estimate of it is available.…”