IEEE Circuits and Systems Magazine - Q1 2020 - 45

Hardware-oriented adaptation techniques that are
actually able to make the difference when
coming to real-world implementations.
As a common starting point for all methods, note that,
if the columns of B are normalized to unit length, the very
same definition of G implies that n (B) is the maximum of
the magnitudes of the off-diagonal entries in G. The prototype optimization problem that is being solved is
min

G ! G, F ! F

G-F

(3)

#

where the $ # norm can be either the sup norm $ 3 or
the Frobenius norm $ F and the matrix set F is suitably
defined in every variant of (3). As far as G is concerned,
we need to limit the search to symmetric, positive semidefinite, low-rank matrices with a unit diagonal, i.e.,
G = " G ; G < = G / G * 0 / rank ^G h = m / diag ^G h = 1 , (4)
<

Since G is symmetric, we have G = Q G K G K G Q G< .
If KlG is the m # m upper left submatrix of K G containing the m non-zero eigenvalues and QlG is the d # m
submatrix of Q G containing the m leftmost columns,
<
then we also have G = QlG KlG KlG (QlG) < implying
<
<
B = KlG (QlG) . With this, we may finally set A = BD +.
As far as G and F are convex sets defined as the intersection of elementary convex sets, the key technique
for solving (3) is a mix of projected gradient descent [17]
or shrinking in which the projection on G and F is computed by the method of alternating projections [18].
What follows is a brief overview of the proposals that
use the above setting, each of them labeled with the prefix
coh- followed by the initial of one of the proposing authors.
1) The "coh-S" Method in [19]
The method is equivalent to set $ # = $ F and F = {I d},
with I d the d # d identity matrix, thus simply pursuing
the reduction of all off-diagonal entries of G.
As noted in [20], pushing G towards I d can be also
interpreted, under suitable assumptions on the signal to
acquire, in terms of minimization of the average squared
error committed by an oracle estimator of p that knows
in advance which are the non-zero entries.
2) The "coh-C" Method in [21]
The author notices that when d 2 n the dictionary is redundant and this implies some coherence between the
columns of D. Since vectors forming a small angle get
projected into vectors forming a small angle, such a coherence is imported in AD whatever the A.
FIRST QUARTER 2020

Hence, instead of trying to reduce cross correlation,
it is more sensible to make G as close as possible to the
Gram matrix of the dictionary alone D < D. Hence, the
method sets F = " D < D , and considers both $ # = $ F
and $ # = $ 3 .
3) The "coh-X" Method in [22]
The authors note that pushing G towards I d is not completely justified by the objective of making the columns
of B as distinguishable as possible. In fact, assuming
that distinguishable can be interpreted as orthogonal, a
set of d vectors in R m that are as orthogonal as possible
is a Grassmannian frame (GF) [23].
Then it would be convenient to define F as the set
of all the possible Gram matrices corresponding to a
GF. From [23] we know that, if all the columns are normalized to unit length, the absolute value of the scalar
product of every pair of vectors in a GF matches n min .
Hence, the corresponding Gram matrix F is such that
Fj, k = ! n min for every j ! k. Regrettably the set of such
matrices is not convex and [22] relaxes it to one of its
convex supersets. In particular it considers the set of
symmetric, unit diagonal matrices, whose off-diagonal entries have a magnitude not larger than n min, i.e.,
F = " F | F < = F / F * 0 / diag (F ) = 1 / ; Fj, k ; # n min 6j ! k ,.
4) The "coh-B" Method in [24]
This method puts an even stronger emphasis on framebased design by considering Equiangular Tight Frames
(ETFs) [25], [26]. Tightness means that B < y 22 = a y 22
for some a 2 0. Equiangularity implies a = d/m. Hence,
the SVD of B then becomes B = U B 6 d/m I m 0@ V 


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