Mathematics in Independent Component Analysis

1.7. Outlook 51
noise
data signal
independent structured models
Figure 1.28: Future analysis framework.
of the search space Gl(n) of all invertible (n × n)-matrices, Amari was able to considerably improve
search performance and accuracy thus providing an equivariant ICA algorithm. In Theis
(2005b), we gave an overview of various gradient calculations on Gl(n) and presented generalizations
to over- and undercomplete cases, realized by a semidirect product. These ideas were used
in Squartini and Theis (2006), where we defined some alternative Riemannian metrics on the
parameter space of non-square matrices, corresponding to various translations defined therein.
Such metrics allowed us to derive novel, efficient learning rules for two ICA based algorithms
for overdetermined blind source separation.
In Meyer-Baese et al. (2006), we studied optimization and statistical learning on a neural
network that self-organized to solve a BSS problem. The resulting online learning solution used
the nonstationarity of the sources to achieve the separation. For this, we divided the problem
into two learning problems one of which is solved by an anti-Hebbian learning and the other by
an Hebbian learning process. The stability of related networks is discussed in Meyer-Bäse et al.
(2006).
A major application of unsupervised learning in neuroscience lies in the analysis of functional
MRI data sets. A problem we faced during the course of our analyses was the efficient storage
and retrieval of many, large-dimensional spatiotemporal data sets. Although some dimension
reduction or region-of-interest selection may be performed beforehand to reduce the sample
size (Keck et al., 2006), we wanted to compress the data as well as possible without loosing
information. For this, we proposed a novel lossless compression method named FTTcoder (Theis
and Tanaka, 2005) for the compression of images and 3d sequences collected during a typical
fMRI experiment. The large data sets involved in this popular medical application necessitated
novel compression algorithms to take into account the structure of the recorded data as well as
the experimental conditions, which include the 4d recordings, the used stimulus protocol and
marked regions of interest (ROI). For this, we used simple temporal transformations and entropy
coding with context modeling to encode the 4d scans after preprocessing with the ROI masking.
The compression algorithm as well as the fMRI toolbox and the algorithms for spatiotemporal
and subspace BSS are all available online at http://fabian.theis.name.