Learning with Transformation Invariant Kernels

Walder, C; Chapelle, O

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Learning with Transformation Invariant Kernels

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Walder, C
Department Human Perception, Cognition and Action, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;
Project group: Cognitive Engineering, Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

Walder, C., & Chapelle, O. (2008). Learning with Transformation Invariant Kernels. In C. Platt, D. Koller, Y. Singer, & S. Roweis (Eds.), Advances in Neural Information Processing Systems 20: 21st Annual Conference on Neural Information Processing Systems 2007 (pp. 1561-1568). Red Hook, NY, USA: Curran.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-C749-7

Abstract

This paper considers kernels invariant to translation, rotation and dilation. We show that no non-trivial positive definite (p.d.) kernels exist which are radial and
dilation invariant, only conditionally positive definite (c.p.d.) ones. Accordingly, we discuss the c.p.d. case and provide some novel analysis, including an elementary derivation of a c.p.d. representer theorem. On the practical side, we give a support vector machine (s.v.m.) algorithm for arbitrary c.p.d. kernels. For the thinplate
kernel this leads to a classifier with only one parameter (the amount of regularisation), which we demonstrate to be as effective as an s.v.m. with the Gaussian kernel, even though the Gaussian involves a second parameter (the length scale).