Abstract
In this article we investigate the field of Hilbertian metrics on probability measures. Since they are very versatile and can therefore be applied in various problems they are of great interest in kernel methods. Quit recently Topsøe and Fuglede introduced a family of Hilbertian metrics on probability measures. We give basic properties of the Hilbertian metrics of this family and other used metrics in the literature. Then we propose an extension of the considered metrics which incorporates structural information of the probability space into the Hilbertian metric. Finally we compare all proposed metrics in an image and text classification problem using histogram data.
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
Chapelle, O., Haffner, P., Vapnik, V.: SVMs for histogram-based image classification. IEEE Transactions on Neural Networks 10, 1055–1064 (1999)
Fuglede, B.: Spirals in Hilbert space. With an application in information theory. To appear in Expositiones Mathematicae (2004)
Hein, M., Bousquet, O.: Maximal margin classification for metric spaces. In: 16th Annual Conference on Learning Theory, COLT (2003)
Jebara, T., Kondor, R.: Bhattacharyya and expected likelihood kernels. In: 16th Annual Conference on Learning Theory, COLT (2003)
Lafferty, J., Lebanon, G.: Diffusion kernels on statistical manifolds. Technical Report CMU-CS-04-101, School of Computer Science, Carnegie Mellon University, Pittsburgh (2004)
Schoenberg, J.: Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44, 522–536 (1938)
Schölkopf, B.: The kernel trick for distances. In: NIPS, vol. 13 (2000)
Topsøe, F.: Some inequalities for information divergence and related measures of discrimination. IEEE Trans. Inform. Th. 46, 1602–1609 (2000)
Topsøe, F.: Jenson-shannon divergence and norm-based measures of discrimination and variation. Preprint (2003)
Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree. Adv. Comp. Math. 4, 389–396 (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hein, M., Lal, T.N., Bousquet, O. (2004). Hilbertian Metrics on Probability Measures and Their Application in SVM’s. In: Rasmussen, C.E., Bülthoff, H.H., Schölkopf, B., Giese, M.A. (eds) Pattern Recognition. DAGM 2004. Lecture Notes in Computer Science, vol 3175. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-28649-3_33
Download citation
DOI: https://doi.org/10.1007/978-3-540-28649-3_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22945-2
Online ISBN: 978-3-540-28649-3
eBook Packages: Springer Book Archive