IEEE Signal Processing - July 2018 - 57
sensing allows underdetermined estimation (from sub-Nyquist
samples) under the assumption that the signal to be estimated
allows sparse representation over suitable basis [5]. However, in
many practical applications, the goal is to infer the parameter of
interest from the second-order statistics (or power spectrum) of the
data (which often acts as a sufficient statistic). Popular examples
include audio and speech processing, communication, radar and
sonar signal processing, radio astronomy, seismology, and superresolution imaging.
A main theme of this article is to establish that in such correlation-driven estimation problems, it is possible to estimate
parameters of interest from compressive measurements, without
requiring the signal to have a sparse representation [6]. What
enables compression is the fact that 1) the second-order statistics or
correlation of the signal of interest exhibit special structures (such
as Toeplitz) due to physics of the problem, and 2) one only needs
to reconstruct the second-order statistics, instead of the entire time
series. These insights have led to an emerging body of work-
compressive covariance sensing (CCS) [6]-[10]. The main idea
in these problems is to design appropriate samplers (in temporal
and spatial domains) that can maximally exploit the structure of
the covariance matrix and enable compression without sparsity.
For harmonic retrieval and direction of arrival (DOA) estimation problems that frequently arise in radar signal processing, the
design of such samplers is governed by the so-called difference
sets, and nonuniform arrays such as nested and coprime arrays
are known to be order-wise optimal for covariance compression
[11], [12]. These designs can be further generalized to compressively sample the power spectrum of any wide-sense stationary
(WSS) signal [6].
In addition to a structured covariance matrix then the signal
of interest also exhibits sparsity over suitable basis, the correlation structure of the signal can be utilized to attain compression
factors that cannot be obtained by exploiting sparsity alone. Such
signal models are unified under the umbrella of sparse Bayesian
learning (SBL) [13]-[15] and Bayesian compressed sensing [16].
The problem of sparse signal recovery is cast in the Bayesian setting by imposing certain sparse priors on the correlation matrix
of the signal of interest. The sparsity in these models is equivalently controlled by certain hyperparameters that represent the
signal power. The main intuition is that the problem of recovering the sparse support (which is a detection problem) can be cast
as a problem of estimating these hyperparameters. In this article,
we show that, in such cases, the nonzero parameters (or model
order) can potentially exceed the measurement dimension [10], a
feat that cannot be achieved by exploiting sparsity alone. The key
idea is to design efficient samplers that can map the underdetermined problem into a suitable higher-dimensional space (which
is derived from the correlation of the data) and exploit low-rank
structures in this space.
A previous survey article [6] focuses on the role of structured
samplers (including those inspired from difference sets) and
least-squares-based reconstruction of the covariance matrix under a variety of stationary signal models. In this article, we go
beyond the question of compressing and reconstructing covariance matrices, and understand how exploitation of correlation
priors can fundamentally improve parameter identifiability by
clever design of samplers. In this context, we review recent advancements toward the analysis of such underdetermined estimation problems using new results on CRBs, and establish stability
guarantees of popular convex and nonconvex algorithms. These
results will also reveal important connections between correlation-aware techniques and a rich line of work on SBL, which is a
powerful Bayesian tool for sparse signal reconstruction utilizing
correlation of the data.
Notation
Throughout this article, matrices are represented by bold uppercase letters, vectors by bold lowercase letters. The symbol x i
denotes the ith entry of a vector x. The notation A S (respectively, x S ) represents the submatrix (respectively, subvector) of A
(respectively, x) whose columns (elements) are indexed by the set
of integers S. The symbols 9 and , represent the Khatri-Rao
and Kronecker products, respectively.
Role of samplers in correlation-aware
low-rank inverse problems
To illustrate the role of samplers in inverse problems that utilize
correlation of the data, we consider the following model that collects a set of L independent measurements:
y 6 l @ = Sx 6 l @ + w 6 l @, l = 1, 2, f L.
(1)
Here y [l] ! C M denotes the lth measurement vector, S ! C M # N
is a sampling matrix, x [l] ! C N is the unknown signal of interest, and w [l] ! C M is the additive noise. We will assume all
random variables to be zero-mean, unless otherwise stated. In
many inverse problems that arise in signal processing and imaging, the covariance matrix R xx ^Hh = E ^x [l] x [l] H h of x [l] is
characterized by a physically meaningful parameter of interest
H ! R D . Often in such cases, R xx (H) is also a low-rank matrix
whose rank is proportional to the dimension D of the parameter
vector H (D 1 N). For source localization problems studied in
this article, D is typically twice the rank of R xx (H) . The low
rank of R xx (H) can be exploited to compress the signal x [l] and
acquire compressive measurements y [l] of dimension M 1 N
using a suitable sampling matrix S. The central goals in such
inverse problems are twofold.
■ How does one design a sampling matrix S that maximally
compresses the data while allowing the parameter H to
remain identifiable?
■ How does one estimate H from these compressive measurements y [l], l = 1, 2, f, L with provable guarantees?
Estimation of H often relies upon exploiting the unique algebraic structure and low-rank of the signal covariance matrix
R xx (H), leading to the notion of a correlation-aware low-rank
inverse problem. This problem has been of significant interest
in recent times, owing to its intimate connections with CCS and
compressive power spectrum estimation [6], [7], [17]-[20], underdetermined source localization problems with nonuniform arrays
[11], [21]-[27], and SBL [13]-[16]. In all these problems, design
of the sampling matrix S plays a critical role in determining the
IEEE SIgnal ProcESSIng MagazInE
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July 2018
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57
Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018
Contents
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