Abstract
In this paper we present a method for functional principal component analysis based on the Oja-learning and neural gas vector quantizer. However, instead of the Euclidean inner product the Sobolev counterpart is applied, which takes the derivatives of the functional data into account and, therefore, uses information contained in the functional shape of the data into account. We investigate the theoretical foundations of the algorithm for convergence and stability and give exemplary applications.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Kantorowitsch, I., Akilow, G.: Funktionalanalysis in normierten Räumen, 2nd revised edn. Akademie-Verlag, Berlin (1978)
Kolmogorov, A., Fomin, S.: Reelle Funktionen und Funktionalanalysis. VEB Deutscher Verlag der Wissenschaften, Berlin (1975)
Kushner, H., Clark, D.: Stochastic Appproximation Methods for Constrained and Unconstrained Systems. Springer, New York (1978)
Labusch, K., Barth, E., Martinetz, T.: Learning data representations with sparse coding neural gas. In: Verleysen, M. (ed.) Proceedings of the European Symposium on Artificial Neural Networks ESANN, pp. 233–238. D-Side Publications (2008)
Landgrebe, D.: Signal Theory Methods in Multispectral Remote Sensing. Wiley, Hoboken (2003)
Lee, J., Verleysen, M.: Generalization of the l p norm for time series and its application to self-organizing maps. In: Cottrell, M. (ed.) Proc. of Workshop on Self-Organizing Maps (WSOM) 2005, Paris, Sorbonne, pp. 733–740 (2005)
Martinetz, T.M., Berkovich, S.G., Schulten, K.J.: ’Neural-gas’ network for vector quantization and its application to time-series prediction. IEEE Trans. on Neural Networks 4(4), 558–569 (1993)
Oja, E.: Neural networks, principle components and suspaces. International Journal of Neural Systems 1, 61–68 (1989)
Oja, E.: Nonlinear pca: Algorithms and applications. In: Proc. of the World Congress on Neural Networks Portland, Portland, pp. 396–400 (1993)
Olshausen, B., Finch, D.: Emergnece of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381, 607–609 (1996)
Ramsay, J., Silverman, B.: Functional Data Analysis, 2nd edn. Springer Science+Media, New York (2006)
Rossi, F., Delannay, N., Conan-Gueza, B., Verleysen, M.: Representation of functional data in neural networks. Neurocomputing 64, 183–210 (2005)
Sanger, T.: Optimal unsupervised learning in a single-layer linear feedforward neural network. Neural Networks 12, 459–473 (1989)
Triebel, H.: Analysis und mathematische Physik. BSB B.G., 3rd revised edn., Teubner Verlagsgesellschaft, Leipzig (1989)
Villmann, T.: Sobolev metrics for learning of functional data - mathematical and theoretical aspects. Machine Learning Reports, 1(MLR-03-2007), 1–15 (2007) ISSN:1865-3960, http://www.uni-leipzig.de/~compint/mlr/mlr_01_2007.pdf
Villmann, T., Claussen, J.-C.: Magnification control in self-organizing maps and neural gas. Neural Computation 18(2), 446–469 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Villmann, T., Hammer, B. (2009). Functional Principal Component Learning Using Oja’s Method and Sobolev Norms. In: Príncipe, J.C., Miikkulainen, R. (eds) Advances in Self-Organizing Maps. WSOM 2009. Lecture Notes in Computer Science, vol 5629. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02397-2_37
Download citation
DOI: https://doi.org/10.1007/978-3-642-02397-2_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02396-5
Online ISBN: 978-3-642-02397-2
eBook Packages: Computer ScienceComputer Science (R0)