Mathematics in Independent Component Analysis

86 Chapter 3. Signal Processing 84(5):951-956, 2004
the preceding section—here dependence is allowed
between real-valued couples of random variables.
Let k; n ∈ N such that k divides n. We call an
n-dimensional random vector Y k-independent if the
k-dimensional random vectors
⎛ ⎞ ⎛ ⎞
⎜
⎝
Y1
.
Yk
⎟
⎠ ;:::;
⎜
⎝
Yn−k+1
.
Yn
⎟
⎠
are mutually independent. A matrix W ∈ Gl(n; R) is
called a k-multidimensional ICA of an n-dimensional
random vector X if WX is k-independent. If k =1,
this is the same as ordinary ICA.
Obvious indeterminacies are, similar to ordinary
ICA, invertible transforms in Gl(k; R) in each tuple
as well as the fact that the order of the independent
k-tuples is not xed. So, de ne for r; s =1;:::;n=k
the (r; s) sub-k-matrix of W =(wij) tobethek × k
submatrix
(wij) i=rk; :::; rk+k−1
j=sk; :::; sk+k−1
that is the k × k submatrix of W starting at position
(rk; sk). A matrix L ∈ Gl(n; R) is said to be a k-scaling
and permutation matrix if for each r=1;:::;n=kthere
exists precisely one s with the (r; s) sub-k-matrix of
L to be nonzero, and such that this submatrix is in
Gl(k; R), and if for each s=1;:::;n=kthere exists only
one r with the (r; s) sub-k-matrix satisfying the same
condition. Hence, if Y is k-independent, also LY is
k-independent.
Two matrices A and B are said to be k-equivalent,
A ∼k B, if there exists such a k-scaling and permutation
matrix L with A=BL. As stated above, given two
matrices W and V with W −1 ∼k V −1 such that one of
them is a k-multidimensional ICA of a given random
vector, then so is the other. We will show that there are
no more indeterminacies of multidimensional ICA.
As usual multidimensional ICA can solve the multidimensional
BSS problem
X = AS;
where A ∈ Gl(n; R) and S is a k-independent
n-dimensional random vector. Finding the indeterminacies
of multidimensional ICA then shows that
A can be found except for k-equivalence (separability),
because if X = AS and W is a demixing matrix
F.J. Theis / Signal Processing 84 (2004) 951 – 956 955
such that WX is k-independent, then WA ∼k I, so
W−1 ∼k A as desired.
However, for the proof we need one more condition
for A: We call A k-admissible if for each r; s =
1;:::;n=kthe (r; s) sub-k-matrix of A is either invertible
or zero. Note that this is not a strong restriction—
if we randomly choose A with coe cients out of a continuous
distribution, then with probability one we get
a k-admissible matrix, because the non-k-admissible
matrices ⊂ Rn2 lie in a submanifold of dimension
smaller than n2 .
Theorem 5.1 (Separability of multidimensional
BSS): Let A ∈ Gl(n; R) and S a k-independent
n-dimensional random vector having no Gaussian
k-tuple (Srk;:::;Srk+k−1) T . Assume that A is
k-admissible.
If AS is again k-independent, then A is k-equivalent
to the identity.
For the case k = 1 this is linear BSS separability
because every matrix is 1-admissible.
Proof. Denote X := AS. Assume that A k I.
Then there exist indices r1;r2 and s such that
the (r1;s) and the (r2;s) sub-k-matrices of A are
non-zero (hence in Gl(k; R) by k-admissability).
Applying Corollary 3.3 to the two random vectors
(Xr1k;:::;Xr1k+k−1) T and (Xr2k;:::;Xr2k+k−1) T then
shows that (Ssk;:::;Ssk+k−1) T is Gaussian, which is a
contradiction.
Note that we could have used whitening to assume
that A is orthogonal; however there does not seem to
be a direct way to exploit this in order to allow one
fully Gaussian k-tuple, contrary to the complex ICA
case, see Theorem 4.1.
6. Conclusion
Uniqueness and separability results play a central
role in solving BSS problems since they allow algorithms
to apply ICA in order to uniquely (except for
scaling and permutation) nd the original mixing matrices.
We have used a multidimensional version of the
Skitovitch–Darmois theorem in order to calculate the