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Distance-based classification with Lipschitz functions

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http://pubman.mpdl.mpg.de/cone/persons/resource/persons76237

von Luxburg,  U
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons83824

Bousquet,  O
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

von Luxburg, U., & Bousquet, O. (2003). Distance-based classification with Lipschitz functions. In Learning Theory and Kernel Machines, Proceedings of the 16th Annual Conference on Computational Learning Theory (pp. 314-328).


Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-DD9A-8
Abstract
The goal of this article is to develop a framework for large margin classification in metric spaces. We want to find a generalization of linear decision functions for metric spaces and define a corresponding notion of margin such that the decision function separates the training points with a large margin. It will turn out that using Lipschitz functions as decision functions, the inverse of the Lipschitz constant can be interpreted as the size of a margin. In order to construct a clean mathematical setup we isometrically embed the given metric space into a Banach space and the space of Lipschitz functions into its dual space. Our approach leads to a general large margin algorithm for classification in metric spaces. To analyze this algorithm, we first prove a representer theorem. It states that there exists a solution which can be expressed as linear combination of distances to sets of training points. Then we analyze the Rademacher complexity of some Lipschitz function classes. The generality of the Lipschitz approach can be seen from the fact that several well-known algorithms are special cases of the Lipschitz algorithm, among them the support vector machine, the linear programming machine, and the 1-nearest neighbor classifier.