Digital Signal Processing Reference
In-Depth Information
block of size
m
k
,
f
(
A
), when restricted to the block, is maximal over
A
∈
O
(
m
k
), which we denote as
block-optimal
. The proof is given in
[251].
In the case of
k
-ISA, where
m
=(
k,...,k
), we used this result
to propose an explicit algorithm [249]. Consider the BSS model from
equation (4.1). As usual, by preprocessing we may assume whitened
observations
x
,so
A
is orthogonal. For the density
p
s
of the sources, we
therefore get
p
s
(
s
0
)=
p
x
(
As
0
). Its Hessian transforms like a 2-tensor,
which locally at
s
0
(see section 4.2) guarantees
H
ln p
s
(
s
0
)=
H
ln p
x
◦
A
(
s
0
)=
AH
ln p
x
(
As
0
)
A
.
(5.7)
The sources
s
(
t
) are assumed to be
k
-independent, so
p
s
factorizes into
r
groups each depending on
k
separate variables Thus ln
p
s
is a sum
of functions depending on
k
separate variables, and hence
H
ln p
s
(
s
0
)is
k
-block diagonal. Hessian ISA now simply uses the block-diagonality
structure from equation (5.7) and performs JBD of estimates of a set
of Hessians
H
ln p
s
(
s
i
) evaluated at different sampling points
s
i
.This
corresponds to using the HessianICA source condition from table 5.1.
Other source conditions, such as contracted quadricovariance matrices
[46] can also be used in this extended framework [251].
Unknown group structure: General ISA
A serious drawback of
k
-ISA (and hence of ICA) lies in the fact that
the requirement of fixed group size
k
does not allow us to apply this
analysis to an arbitrary random vector. Indeed, theoretically speaking,
it may be applied only to random vectors following the
k
-ISA blind
source separation model, which means that they have to be mixtures of
a random vector that consists of independent groups of size
k
.Ifthis
is the case, uniqueness up to permutation and scaling holds according
to theorem 5.1. However, if
k
-ISA is applied to any random vector, a
decomposition into groups that are only “as independent as possible”
cannot be unique, and depends on the contrast and the algorithm. In
the literature, ICA is often applied to find representations fulfilling the
independence condition only as well as possible. However, care has to be
taken; the strong uniqueness result is not valid anymore, and the results
may depend on the algorithm as illustrated in figure 5.3.
In contrast to ICA and
k
-ISA, we do not want to fix the size of the
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