IEEE Signal Processing - July 2018 - 63

[14]. In compressed sensing, given the multiple measurement vectors (MMVs) y [l], l = 1, 2, f, L, the support S is
usually recovered by first reconstructing the sparse vectors
x [l], l = 1, 2, f, L and then finding their common support.
However, SBL follows a different approach and recovers S by
estimating the power c. With regards to the size of the recoverable support, there is a significant difference between these
two approaches. It is well known that for MMV models, the
joint support S of {x [l]} lL= 1 is uniquely identifiable if [48]
S 1

Kruskal-rank ^Sh + rank ^Xh
,
2

(11)

where X = [x [1], x [2], f, x [L]]. Since Kruskal-rank (S) 1
M and rank (X) # S , this bound implies that the size of the
support cannot exceed M, i.e., S 1 M. Most existing compressed sensing algorithms that recover the common support
from MMV models obey this bound [49]-[54].

Khatri-Rao product of measurement matrix
The bound (11) is pessimistic for SBL since it ignores the correlation structure of the sparse signals x [l]. Since SBL detects
support by estimating the hyperparameter c, the maximum
recoverable size of S is related to the identifiability of the
parameter c. To illustrate this, suppose we know an upper
bound on the size of S, i.e.,
S # s max .

s max) lead to the same distribution of the data. In this case,
it can be easily shown that no support detector will be able
to distinguish between S 1 and S 2 from the measurements
y [l], l = 1, 2, f, L for any value of L. Such nonidentifiability occurs if there exist two distinct hyperparameters c 1
and c 2, c 1 ! c 2 with supp (c 1) = S 1 and supp (c 2) = S 2
such that
SC 1 S H = SC 2 S H .
The above condition is equivalent to
^ S ) 9 S h^c 1 - c 2 h = 0.

(14)

2

The matrix S ) 9 S ! C M # N is called the Khatri-Rao product [55] of S. Each column of S ) 9 S is a Kronecker product
of corresponding columns of S ) and S, i.e.,
S ) 9 S = [s )1 7 s 1, s )2 7 s 2, f, s )N 7 s N ] .

(15)

It is the Kruskal-rank of this Khatri-Rao product S ) 9 S (and
not of S) that determines the maximum size s max of the recoverable support in SBL. Since the number of nonzero
elements in c 1 - c 2 can be at most 2s max, it follows that
nonidentifiability will not occur as long as
Kruskal-rank (S ) 9 S) 2 2s max .

(12)

The desired support is said to be nonidentifiable, if two distinct supports S 1 ! S 2, (each with a size no larger than

(13)

(16)

Figure 5 pictorially depicts the distinction between conventional support recovery and correlation-aware support

Conventional MMV Support Recovery

Correlation-Aware Support Recovery
M2 × N

M×N

Y=

Vec(Ryy) =
A

x1x2 . . xL

Same
Support S

Common
Support S
Unique Sparsest Solution If

S <

Unique Sparsest Solution If

Kruskal-Rank (S) + Rank (X)

Kruskal-Rank (S∗

S <

2

S)

2

Figure 5. The conventional approach to support recovery in MMV models versus correlation-aware support recovery. For the latter, the Kruskal rank of
the Khatri-Rao product of the measurement matrix determines the maximum size of recoverable support.
IEEE Signal Processing Magazine

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July 2018

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63
Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018
