IEEE Signal Processing - July 2018 - 64

recovery in MMV models. By exploiting the nonnegativity of
c, the bound (16) can be further improved to
Kruskal-rank (S ) 9 S) 2 s max

(17)

for certain well-designed measurement matrices S whose
Khatri-Rao products have Vandermonde structure [56].

Deterministic and random measurement matrices
The Khatri-Rao product actually generalizes and extends the
notion of difference sets, beyond measurement matrices
encountered in source localization type of problems. SBL can
be used for grid-based DOA estimation, where the angle
space is divided into N candidate directions. In this case, S
has the structure
S m, n = e j2rd m n/N ,

(18)

where d m denotes the location of the mth sensor. The elements of the Khatri-Rao product S ) 9 S are given by
[S ) 9 S] (m - 1) M + ml, n = e j2r (d m - d ml) n/N .

(19)

The rows of S ) 9 S are therefore characterized by the difference set S diff introduced previously in (3). The number of distinct elements in S diff therefore fundamentally limits the
Kruskal rank of S ) 9 S and serves as an upper bound for
s max . As shown previously, the Kruskal rank of ULAs with
M sensors is 2M - 1, whereas that of a nested array is
M 2 /2 + M - 1. This further justifies that using nested arrays,
size of identifiable supports can be as large as s max = O (M 2).
Apart from the aforementioned deterministic classes of
measurement matrices, random measurement matrices also
exhibit a very large Kruskal rank with probability one. Let
S ! R M # N be a random matrix whose elements are drawn
independently from a continuous distribution over R MN .
Assuming N 2 (M 2 + M) /2, the following holds with probability one [10]:
2
Kruskal-rank (S ) 9 S) = M + M .
2

(20)

Hence, almost all choices of i.i.d. random matrices (irrespective of their distribution) will have Kruskal rank as large
as O (M 2) and potentially allow recovery of supports of
size O (M 2).
Although our discussion so far has been primarily focused
on compressing Toeplitz-structured covariance matrices, the
idea can be potentially extended to compress covariance matrices with more general structures. For example, in the emerging field of graph signal processing, the concept of difference
sets and sparse rulers can be extended to compressively sample
families of second-order stationary graph signals [57]-[59]. In
a related class of problems, the role of structured samplers for
optimally compressing sparse covariance matrices has been
studied in using measurements matrices representing adjacency matrices of random bipartite graphs [60].
64

Algorithms and fundamental limits for correlationaware compressive estimation
So far, we discussed optimal compression techniques that enable
exact identification of parameters of interest from compressive
sketches of the signal covariance matrix. However, in practice,
the compressed sketch is only estimated from a finite number
t yy = (1/L) R lL= 1 y [l] y H [l]. One can model this
of samples as R
estimated data covariance matrix as
t yy = SR xx S H + v 2w I + E L,
R

(21)

where E L represents errors due to finite sample averaging. Pract yy need to be cognitical algorithms for estimating H from R
zant of the mismatch E L due to finite samples. Two questions
are of importance in this regard:
1) How to estimate the signal covariance matrix R xx from the
t yy? In particular, how can the
estimated compressive sketch R
rank and algebraic structure of R xx be utilized to develop
theoretical guarantees?
2) How accurately can we estimate the desired parameter H
from the estimate of R xx (H) ?

Algorithms for compressive covariance estimation
There is a rich literature on the problem of recovering low-rank
matrices from compressive or incomplete observations (see [1]
for an up-to-date survey). While earlier algorithms aimed to
relax the nonconvex rank constraint using a convex surrogate
(nuclear norm), recent approaches attempt to solve the nonconvex problem using suitable initialization techniques [61]-
[65]. However, when the underlying low-rank matrix is also a
covariance matrix, it has additional properties, such as positive (semi) definiteness, and/or Toeplitz structure. These properties can be exploited to develop more specialized algorithms
with stronger performance guarantees. Indeed, estimating the
high-dimensional covariance matrix from corrupted (possibly
due to finite samples) measurements has received significant
research interest in recent times, especially due to its connections to covariance sampling using the so-called quadratic
measurement model [9].

Random rank-1 measurements and covariance estimation
The quadratic measurement model is inspired from that of
phase retrieval [66], where one collects measurements of
the form
z i = a iH xx H a i + n i, i = 1, 2, f, M.

(22)

Here, x ! C N is the desired signal of interest and n i models
the noise and/or error. In absence of noise, z i is a quadratic
function of x, justifying the term quadratic measurement
model. When R xx is a rank-1 covariance matrix, (and is,
therefore, of the form R xx = xx H), the measurements z i rept yy . The model
resent the diagonal elements of the sketch R
can be extended to represent sketches of any covariance
matrix R xx (with arbitrary rank r) acquired as [9]

IEEE Signal Processing Magazine

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July 2018

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Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018

Contents
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