IEEE Signal Processing - July 2018 - 62
X
Ryy
Compress
Y
Size O (r 1/2)
Size N
Tr +1
Predict ω , c
Subspace
i i
i = 1, 2, .. r Algorithms
Recover
TN
Figure 4. The GNS-based compression and reconstruction technique
for WSS signals with low-rank Toeplitz covariance matrices. The GNS
produces a sketch of size M # M, where M = O ( r ). The 2r parameters
associated with the low-rank PSD Toeplitz matrix can be exactly reconstructed from this sketch (in absence of noise) and the remaining entries
of the Toeplitz matrix can be predicted.
elements of R xx in a systematic way into the sketch R yy, which
has a total of O (N ) elements.
Since a real-valued symmetric N # N Toeplitz matrix contains only N distinct entries, the size of the order-wise optimal
sketch should be O ( N ) # O ( N ). The compression factor
attained by GNS is therefore order-wise optimal (the same as the
sparse ruler). However, GNS has the added advantage of having
an analytic closed form, which makes it computationally tractable [43].
Optimal compression of low-rank Toeplitz matrices
Effective compression of low-rank positive semidefinite (PSD)
Toeplitz matrices require careful exploitation of 1) the redundancies present due to Toeplitz structure, as well as 2) its low
rank. This can be achieved by invoking Carathéodory's lemma,
which provides a powerful parametric representation for lowrank PSD Toeplitz matrices. A PSD Toeplitz matrix T ! C N # N
with rank r 1 N has the following decomposition [44]:
T = V N DV HN ,
(7)
where V N ! C N # r = [v N ( f1), v N ( f2), f, v N ( fr)] is a Vandermonde matrix, with
6v N ^ fi h@k = e j2rfi (k - 1) fi ! ^^- 1 2 h, ^1 2 h@, 1 # k # N. (8)
correspond to line spectrum processes and are uniquely determined by the frequencies and their powers. This representation also shows how to optimally compress such matrices
using the proposed GNS matrix. Notice that the (r + 1) #
(r + 1) principal minor of T (denoted Tr + 1) is also a rankdeficient PSD Toeplitz matrix sharing the same parametric
representation characterized by ( fi, d i), i = 1, 2, f, r [45].
Owing to the Vandermonde structure, any subspace-based
harmonic retrieval algorithm (such as MUSIC) can exactly
recover the frequencies fi, i = 1, 2, f r (and subsequently,
the amplitudes d i by inverting a linear system of equations). Hence, if we only retain a sketch of Tr + 1, it is possible
to perfectly recover the parameters fi, d i, and hence compute
any remaining entry of T. Since Tr + 1 is also Toeplitz structured, one can simply design a GNS to produce a compressive
sketch of size O ^ r h # O ^ r h. The proposed compression
and reconstruction of WSS signals with low-rank Toeplitz
covariance matrix is schematically depicted in Figure 4
[45]. In the "Algorithms and Fundamental Limits for Correlation-Aware Low-Rank Estimation" section, we compare
the performance of GNS-based low-rank Toeplitz covariance compression with random sampling-based approaches
from [9].
Role of sampling in SBL and correlation-aware
sparse estimation
The sampling strategies discussed so far showed how rank
and correlation structure can be exploited to estimate parameter H from the covariance matrices of optimally compressed
measurements. These ideas can actually be generalized and
applied to a broader class of problems in sparse signal processing, unified under the umbrella of SBL and correlation-aware sparse estimation [10], [13]-[16], [46], [47]. The
measurement model for SBL can be described using the general model (1) by imposing both sparsity and correlation constraints on the signal x [l]. In particular, x [l], l = 1, 2, fL are
assumed to be i.i.d. Gaussian vectors distributed as
x [l] ~N (0, C),
where C ! R N # N is a diagonal matrix with diagonal entries
given by c = [c 1, c 2, f, c N ]. Furthermore, c is assumed to
be a sparse vector with s 1 N nonzero elements. Hence, the
covariance matrix C is low rank, a direct consequence of the
sparsity of c. Let S = supp (c). The sparsity of c directly
controls the sparsity of x [l], i.e., S also represents the common support of the vectors x [l], l = 1, 2, f, L. The measurements y [l] are distributed as
y 6 l @ ~N ^0, SCS H + v 2w Ih .
r#r
The matrix D ! R
is diagonal with positive entries
{d 1, d 2, f, d r}. The representation tells us that a low-rank PSD
Toeplitz matrix is completely specified by 2r quantities:
fi, d i, i = 1, 2, f, r. The frequencies fi, i = 1, 2, f, r represent
spectral lines or harmonics and d i represent the corresponding
amplitudes. Hence, low-rank PSD Toeplitz matrices naturally
62
(9)
(10)
The goal in SBL and correlation-aware sparse estimation
[10], [14], [15] is to determine the (common) support S of the
vectors x [l] by using its correlation structure. SBL achieves
this by estimating the hyperparameter c from the measurements y [l], and its support serves as an estimate of S [13],
IEEE Signal Processing Magazine
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July 2018
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