IEEE Signal Processing - July 2018 - 68

3) Assuming equal source powers, the CRB shows a saturation
effect with respect to the SNR pr when D 2 M. This is commensurate with the asymptotic behavior (with respect to SNR)
of the estimates obtained from SS-MUSIC algorithm. Under
certain mild conditions, it can be shown that the CRB converges to zero as pr " 3 for D 1 M, whereas for D 2 M, it
stays strictly bounded away from zero.

Support recovery and SBL
The conceptual insights from source localization in the regime
D 2 M can be directly extended to support recovery using SBL.
As already discussed in the section "Role of Samplers in Correlation-Aware Low-Rank Inverse Problems," if an upper bound
^s max h on the size of the sparse support is known, then the joint
support S can be uniquely identified even when its size exceeds
M (dimension of individual measurement vectors). This precisely happens since we recover the support from the hyperparameter c that captures the correlation structure of the unknown
signal X.

Support recovery via estimation

In SBL, the recovered support St is obtained as St = supp (ct ),
where ct is the solution to the following ML problem [14]:
min
log SCS H + v 2 I + 1 Trace ^(SCS H + v 2 I) - 1 YY H h .
c, v
L
(37)
Analysis of (37), which is a nonconvex problem, is challenging in
the most general setting. However, the results in [13] offer significant insights into the global and local minima of (37), when
D 1 M. In particular, the following are true when D 1 M [13]:
1) In absence of noise (i.e., w [l] = 0h, the global minimum of
(37) is attained at the ct = c 0, vt = 0.
2) Regardless of whether noise is present, the local minima of
(37) have no more than M nonzero entries.
Analysis of the local and global minima of (37) when D 2 M
is an open problem. It is imperative to ask if, at all, the hyperparameter c can be uniquely identified, asymptotically as L " 3.
A partial answer to this question was provided in [82]: the
authors showed that, for finite L, if the rows of X are orthogonal to each other, then the global minimum of (37) is indeed
attained at the true hyperparameter c. This happens even when
D 2 M, provided the Kruskal-rank condition is satisfied. This
observation, which is true under certain deterministic orthogonality conditions on X, can be generalized by considering the
stochastic Gaussian model for X and deriving the CRB for estimating c under this model. An important observation in this
case is that for solving (37), SBL does not impose any sparsity
constraint on the parameter c. Hence, the appropriate CRB in
this case is derived by simply treating c as the unknown realvalued parameter (without specific constraints) that characterize
the distribution of the data y [l] . A deeper understanding of the
structure of the FIM for D 2 M was offered in [83] that alluded
to the possibility of recovering supports of size D 2 M. To see
this, assuming v w is known, the FIM J can be shown to have
the following form:
68

J = L (S * 9 S) H (R yy- T 7 R yy- 1) (S * 9 S) .

(38)

Since R yy is always full rank, the FIM is nonsingular as long as
S * 9 S has full column rank (i.e., rank N ). From our previous
discussion on the Kruskal rank of Khatri-Rao products of random matrices, we know that almost all choices of S will produce
S * 9 S with full-column rank N, provided N # ^ M 2 + M h /2.
In this regime, the CRB for c will exist regardless of whether
D 1 M or D 2 M, and will asymptotically converge to zero as
the number of temporal measurements L " 3.
However, the same conclusion cannot be drawn as we increase
the dimension M of each measurement. In this case, it can be
shown that the CRB for estimating c i, i = 1, 2, f, N exhibits
distinct behavior depending on whether c i is zero or nonzero (i.e.,
whether i ! S or i ! S ch [84]. Keeping L fixed at L = 1, as
M " 3, the CRB of c i saturates at
[J - 1] i, i $ c 2i , i ! S .

(39)

Such saturation happens for both D 1 M and D 2 M, regardless
of the structure of the measurement matrix. Hence, it is not possible to consistently estimate the nonzero entries of c i by arbitrarily
increasing M. On the other hand, the CRB corresponding to the
zero entries of c may or may not exhibit such a saturation effect
depending on the dictionary. For example, if the columns of S are
normalized, i.e., s i 2 = 1, 1 # i # N, then the CRB for the zero
elements remain bounded away from zero as M " 3 [84]. For
unnormalized dictionaries, saturation of the CRB of zero elements
of c is an open question. Summarizing, the CRB for estimating
c exhibits different behavior with respect to the temporal measurements (L) and spatial measurements (M). As long as
N # (M 2 + M) /2, the CRB converges to zero as L " 3. However, as M " 3, the CRB for nonzero elements of c stay bounded away from zero regardless of D 1 M or D 2 M.

Support recovery via detection
While the support of the estimated hyper parameter c serves as
an estimate of S, one can also directly recover the support S
(bypassing the need for estimating any parameter) by considering an appropriate detector that outputs one of ^ DN h candidate
supports. The problem of support recovery in MMV models has
been of great interest in compressed sensing. Sufficient and necessary (information theoretic) bounds on the sample complexity
(as a function of sparsity and ambient dimension) have been
derived for a large class of linear and nonlinear sparse measurement models [85]-[89]. However, these results largely ignore the
statistical model for X. In contrast, the authors in [90] were the
first to propose a multiple hypothesis framework that explicitly
models the correlation of X. In particular, consider the following
Q : = ^ DN h hypothesis, each assuming the data to be generated by
one candidate support

IEEE Signal Processing Magazine

Z
] H 1 : Y~FN (0, R 1 7 I L)
] H 2 : Y~FN (0, R 2 7 I L)
[
,
]h
] H Q : Y~FN (0, R Q 7 I L)
\

|

July 2018

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

Contents
IEEE Signal Processing - July 2018 - Cover1
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IEEE Signal Processing - July 2018 - Cover3
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