Mathematics in Independent Component Analysis

Chapter 4. Neurocomputing 64:223-234, 2005 101
to be of quadrangular shape is sufficient. This gives the idea of a possible
postnonlinear ICA algorithm which minimizes the non-quadrangularity of the
support of the estimated source. As pictured in the graph this can for example
be achieved by minimizing the volume of the mutual differences (i.e. the points
which are in the union but not in the difference). It can easily be seen that this
minimization yields the same solution as minimizing the mutual information.
For more details on such an algorithm we refer to [10, 13, 16].
6 Conclusion
We have presented a new separability result for postnonlinear bounded mixtures
that is based on the analysis of the borders of the mixture density. We
hereby formalize and extend ideas already presented in [10]. We introduce the
notion of absolutely degenerate mixing matrices. Using this we identify the restrictions
of separability and also of algorithms that only use border analysis
for postnonlinearity detection. This also represents a drawback of the algorithms
proposed in [10] and [13], to which we want to refer for experimental
results using border detection in postnonlinear settings.
In future works we will show how to relax the condition of analytic postnonlinearities
to only continuously differentiable functions; also preliminary results
indicate how to generalize these results to complex-valued random vectors and
mixtures. We further plan to extend this model to the case of group ICA [18],
where independence is only assumed among groups of sources. In the linear
case, this has been done in [15] — however the extension to postnonlinearly
mixed sources is yet unclear.
Acknowledgements
We thank the anonymous reviewers for their valuable suggestions, which improved
the original manuscript. FT further would like to thank Christian Jutten
for the helpful discussions during the preparation of this paper. Financial
support by the BMBF in the project ’ModKog’ is gratefully acknowledged.
References
[1] J. Hérault, C. Jutten, Space or time adaptive signal processing by neural
network models, in: J. Denker (Ed.), Neural Networks for Computing.
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