Kernel-based canonical coordinate decomposition of two-channel nonlinear maps

A kernel-based formulation for decomposing nonlinear maps of two data channels into their canonical coordinates is derived. Each data channel is implicitly mapped to a high dimensional feature space defined by a nonlinear kernel. The canonical coordinates of the nonlinear maps are then found by transforming the kernel maps with the eigenvector matrices of a coupled asymmetric generalized eigenvalue problem. This generalized eigenvalue problem is constructed in the explicit space of kernel maps. The measures of linear dependence and coherence between the nonlinear maps of the channels are also presented. These measures may be determined in the kernel domain, without explicit computation of the nonlinear mappings. A numerical example is also presented.

[1]  H. Gish,et al.  Generalized coherence (signal detection) , 1988, ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing.

[2]  Herbert Gish,et al.  A geometric approach to multiple-channel signal detection , 1995, IEEE Trans. Signal Process..

[3]  John K. Thomas,et al.  Wiener filters in canonical coordinates for transform coding, filtering, and quantizing , 1998, IEEE Trans. Signal Process..

[4]  Bernhard Schölkopf,et al.  Nonlinear Component Analysis as a Kernel Eigenvalue Problem , 1998, Neural Computation.

[5]  B. Scholkopf,et al.  Fisher discriminant analysis with kernels , 1999, Neural Networks for Signal Processing IX: Proceedings of the 1999 IEEE Signal Processing Society Workshop (Cat. No.98TH8468).

[6]  Colin Fyfe,et al.  Kernel and Nonlinear Canonical Correlation Analysis , 2000, IJCNN.

[7]  Louis L. Scharf,et al.  Canonical coordinates and the geometry of inference, rate, and capacity , 2000, IEEE Trans. Signal Process..

[8]  Horst Bischof,et al.  Nonlinear Feature Extraction Using Generalized Canonical Correlation Analysis , 2001, ICANN.

[9]  Pedro E. López-de-Teruel,et al.  Nonlinear kernel-based statistical pattern analysis , 2001, IEEE Trans. Neural Networks.

[10]  Bernhard Schölkopf,et al.  Learning with kernels , 2001 .

[11]  Johan A. K. Suykens,et al.  Kernel Canonical Correlation Analysis and Least Squares Support Vector Machines , 2001, ICANN.

[12]  M.R. Azimi-Sadjadi,et al.  Underwater target classification using canonical correlations , 2003, Oceans 2003. Celebrating the Past ... Teaming Toward the Future (IEEE Cat. No.03CH37492).

[13]  Malte Kuss,et al.  The Geometry Of Kernel Canonical Correlation Analysis , 2003 .

[14]  Corinna Cortes,et al.  Support-Vector Networks , 1995, Machine Learning.

[15]  Mahmood R. Azimi-Sadjadi,et al.  Two-channel constrained least squares problems: solutions using power methods and connections with canonical coordinates , 2005, IEEE Transactions on Signal Processing.