Dependent Component Analysis - Biomedical Signal Analysis: Contemporary Methods and Applications - page 150

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 was first intro-
duced by Cardoso [43] using geometrical motivations. His model, as well
as the related but independently proposed factorization of multivariate
function classes [155] are quite general. However, no identifiability results
were presented, and applicability to an arbitrary random vector was un-
clear. 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 [247]. Algorithmic enhancements in this setting have studied been
recently [207]. Similar to [43], Akaho et al. [3] also proposed to employ a
multidimensional-component, maximum-likelihood algorithm, but in the
slightly different context of multimodal component analysis. Moreover,
if the observations contain additional structures such as spatial or tem-
poral structures, these may be used for the multidimensional separation
[126, 276].
Hyvarinen and Hoyer [121] presented a special case of k -ISA by com-
bining it with invariant feature subspace analysis. They model the de-
pendence within a k -tuple explicitly, and are therefore able to propose
more ecient algorithms without having to resort to the problematic
multidimensional density estimation. A related relaxation of the ICA
assumption is given by topographic ICA [122], where dependencies be-
tween all components are assumed and modeled along a topographic
structure (e.g. a two-dimensional grid). However, these two approaches
are not completely blind anymore. Bach and Jordan [13] formulate ISA
as a component clustering problem, which necessitates a model for inter-
cluster independence and intracluster dependence. For the latter, they
propose to use a tree structure as employed by their tree-dependent
component analysis [12]. Together with intercluster independence, this
implies a search for a transformation of the mixtures into a forest (i.e. a
set of disjoint trees). However, the above models are all semiparametric,
and hence not fully blind. In the following, we will review two contribu-
tions, [247] and [251], where no additional structures were necessary for
the separation.