Mathematics in Independent Component Analysis

1.3. Dependent component analysis 19
together with the transformation properties (1.6) of the source conditions yields
Ci( t S) = s S †⊤ Ci(X) s S † and Ci( s S) = t S †⊤ Ci(X ⊤ ) t S †
because ∗ m ≥ n and hence ∗ S ∗ S † = I. By assumption the matrices Ci( ∗ S) are as diagonal as
possible. In order to separate the data, we had to find diagonalizers for both Ci(X) and Ci(X ⊤ )
such that they satisfy the spatiotemporal model (1.8). As the matrices derived from X had to
be diagonalized in terms of both columns and rows, we denoted this by double-sided approximate
joint diagonalization.
In Theis et al. (2007b, 2005a) we showed how to reduce this process to joint diagonalization.
In order to get robust estimates of the source conditions, dimension reduction was essential. For
this we considered the singular value decomposition X, and formulated the algorithm in terms
of the pseudo-orthogonal components of X. Of course, instead of using autocovariance matrices,
other source conditions Ci(.) from table 1.1 can be employed in order to adapt to the separation
problem at hand.
We present an application of the spatiotemporal BSS algorithm to fMRI data using multidimensional
autocovariances in section 1.6.1.
1.3.3 Independent subspace analysis
Another extension of the simple source separation model lies in extracting groups of sources
that are independent of each other, but not within the group. So, multidimensional independent
component analysis or independent subspace analysis (ISA) denotes the task of transforming a
multivariate observed sensor signal such that groups of the transformed signal components are
mutually independent—however dependencies within the groups are still allowed. This allows
for weakening the sometimes too strict assumption of independence in ICA, and has potential
applications in various fields such as ECG, fMRI analysis or convolutive ICA.
Recently we were able to calculate the indeterminacies of group ICA for known and unknown
group structure, which finally enabled us to guarantee successful application of group ICA to
BSS problems. Here, we will shortly review the identifiability result as well as the resulting
algorithm for separating signals into groups of dependent signals. As before, the algorithm is
based on joint (block) diagonalization of sets of matrices generated using one or multiple source
conditions.
Generalizations of the ICA model that are to include dependencies of multiple one-dimensional
components have been studied for quite some time. ISA in the terminology of multidimensional
ICA has first been introduced by Cardoso (1998) using geometrical motivations. His model
as well as the related but independently proposed factorization of multivariate function classes
(Lin, 1998) are quite general, however no identifiability results were presented, and applicability
to an arbitrary random vector was unclear. Later, in the special case of equal group sizes k,
in the following denoted as k-ISA, uniqueness results have been extended from the ICA theory
(Theis, 2004b). Algorithmic enhancements in this setting have been recently studied by Poczos
and Lörincz (2005). Similar to (Cardoso, 1998), Akaho et al. (1999) also proposed to employ
a multidimensional component maximum likelihood algorithm, however in the slightly different
context of multimodal component analysis. Moreover, if the observations contain additional
(1.9)