Abstract
Nonlinear independent components analysis (NICA) is known to be an ill-posed problem when only the independence of the sources are sought. Additional constraints on the distribution of the sources or the structure of the mixing nonlinearity are imposed to achieve a solution that is unique in a suitable sense. In this paper, we present a technique that tackles nonlinear blind source separation (NBSS) as a nonlinear invertible coordinate unfolding problem utilizing a recently developed definition of maximum-likelihood principal curves. The proposition would be applicable most conveniently to independent unimodal source distributions with mixtures that have diminishing second order derivatives along the source axes. Application to multimodal sources would be possible with some modifications that are not discussed in this paper. The ill-posed nature of NBSS is also discussed from a differential geometric perspective in this context.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Hyvarinen, A., Pajunen, P.: Nonlinear Independent Component Analysis: Existence and Uniqueness Results. Neural Networks 12(3), 429–439 (1999)
Almeida, L.: MISEP – Linear and Nonlinear ICA based on Mutual Information. Journal of MachineLearning Research 4, 1297–1318 (2003)
Parra, L., Deco, G., Miesbach, S.: Statistical Independence and Novelty Detection with Information Preserving Nonlinear Maps. Neural Computation 8, 260–269 (1996)
Jutten, C., Karhunen, J.: Advances in Blind Source Separation (BSS) and Independent Component Analysis (ICA) for Nonlinear Mixtures. Int. J. Neural Systems 14(5), 267–292 (2004)
Lee, J.A., Jutten, C., Verleysen, M.: Nonlinear ICA by Using Isometric Dimensionality Reduction. In: Puntonet, C.G., Prieto, A.G. (eds.) ICA 2004. LNCS, vol. 3195, pp. 710–717. Springer, Heidelberg (2004)
Harmeling, S., Ziehe, A., Kawanabe, M., Muller, K.R.: Kernel Based Nonlinear Blind Source Separation. Neural Computation 15, 1089–1124 (2003)
Hastie, T., Stuetzle, W.: Principal Curves. Jour. Am. Statistical Assoc. 84(406), 502–516 (1989)
Tibshirani, R.: Principal Curves Revisited. Statistics and Computation 2, 183–190 (1992)
Sandilya, S., Kulkarni, S.R.: Principal Curves with Bounded Turn. IEEE Trans. on Information Theory 48(10), 2789–2793 (2002)
Kegl, B., Kryzak, A., Linder, T., Zeger, K.: Learning and Design of Principal Curves. IEEE Trans. on PAMI 22(3), 281–297 (2000)
Stanford, D.C., Raftery, A.E.: Finding Curvilinear Features in Spatial Point Patterns: Principal Curve Clustering with Noise. IEEE Trans. on PAMI 22(6), 601–609 (2000)
Chang, K., Grosh, J.: A Unified Model for Probabilistic Principal Surfaces. IEEE Trans. on PAMI 24(1), 59–74 (2002)
Erdogmus, D., Ozertem, U.: Self-Consistent Locally Defined Principal Curves. In: Proceedings of ICASSP 2007, vol. 2, pp. 549–552 (2007)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press and McGraw-Hill (2001)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Erdogmus, D., Ozertem, U. (2008). Nonlinear Coordinate Unfolding Via Principal Curve Projections with Application to Nonlinear BSS. In: Ishikawa, M., Doya, K., Miyamoto, H., Yamakawa, T. (eds) Neural Information Processing. ICONIP 2007. Lecture Notes in Computer Science, vol 4985. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69162-4_51
Download citation
DOI: https://doi.org/10.1007/978-3-540-69162-4_51
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69159-4
Online ISBN: 978-3-540-69162-4
eBook Packages: Computer ScienceComputer Science (R0)