Non-linear independent component analysis with diffusion maps

Amit Singer, Ronald R. Coifman

Research output: Contribution to journalArticle

98 Scopus citations

Abstract

We introduce intrinsic, non-linearly invariant, parameterizations of empirical data, generated by a non-linear transformation of independent variables. This is achieved through anisotropic diffusion kernels on observable data manifolds that approximate a Laplacian on the inaccessible independent variable domain. The key idea is a symmetrized second-order approximation of the unknown distances in the independent variable domain, using the metric distortion induced by the Jacobian of the unknown mapping from variables to data. This distortion is estimated using local principal component analysis. Thus, the non-linear independent component analysis problem is solved whenever the generation of the data enables the estimation of the Jacobian. In particular, we obtain the non-linear independent components of stochastic Itô processes and indicate other possible applications.

Original languageEnglish (US)
Pages (from-to)226-239
Number of pages14
JournalApplied and Computational Harmonic Analysis
Volume25
Issue number2
DOIs
StatePublished - Sep 1 2008
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

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