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# A Spectral Clustering Approach to Underdetermined Post-Nonlinear Blind Source Separation of Sparse Sources

**Steven Van Vaerenbergh, Student Member, IEEE, Ignacio Santamaría, Senior Member, IEEE
**

Dept. of Communications Engineering, University of Cantabria, Spain

E-mail: {steven,nacho}@gtas.dicom.unican.es

Telephone: 34-942201392-13, 34-942201552

Fax: 34-942201488

### Abstract:

This letter proposes a clustering-based approach for solving the
underdetermined (i.e. fewer mixtures than sources) post-nonlinear
blind source separation (PNL BSS) problem when the sources are
sparse. Although various algorithms exist for the underdetermined
BSS problem for sparse sources, as well as for the PNL BSS problem
with as many mixtures as sources, the nonlinear problem in an
underdetermined scenario has not been satisfactorily solved yet.
The method proposed in this work aims at inverting the different
nonlinearities, thus reducing the problem to linear
underdetermined BSS. To this end, first a spectral clustering
technique is applied that clusters the mixture samples into
different sets corresponding to the different sources. Then, the
inverse nonlinearities are estimated using a set of multilayer
perceptrons (MLPs) that are trained by minimizing a specifically
designed cost function. Finally, transforming each mixture by its
corresponding inverse nonlinearity results in a linear
underdetermined BSS problem, which can be solved using any of the
existing methods.

### Index Terms

Blind source separation, underdetermined source separation, post-nonlinear mixtures, sparse sources, spectral clustering,
multilayer perceptrons.

** Next:** Introduction
Steven Van Vaerenbergh

Last modified: 2006-04-05