Diffeomorphic Dimensionality Reduction

Walder, C; Schölkopf, B

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Diffeomorphic Dimensionality Reduction

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Walder, C
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
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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Schölkopf, B
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

Walder, C., & Schölkopf, B. (2009). Diffeomorphic Dimensionality Reduction. In D. Koller, D. Schuurmans, Y. Bengio, & L. Bottou (Eds.), Advances in neural information processing systems 21 (pp. 1713-1720). Red Hook, NY, USA: Curran.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-C479-8

Abstract

This paper introduces a new approach to constructing meaningful lower dimensional representations of sets of data points. We argue that constraining the mapping between the high and low dimensional spaces to be a diffeomorphism is a natural way of ensuring that pairwise distances are approximately preserved. Accordingly we develop an algorithm which diffeomorphically maps the data near to a lower dimensional subspace and then projects onto that subspace. The problem of solving for the mapping is transformed into one of solving for an Eulerian flow field which we compute using ideas from kernel methods. We demonstrate the efficacy of our approach on various real world data sets.