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Proc Natl Acad Sci U S A. 2005 May 24;102(21):7426-31. Epub 2005 May 17.

Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps.

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1
Department of Mathematics, Program in Applied Mathematics, Yale University, New Haven, CT 06510, USA. coifman-ronald@yale.edu

Abstract

We provide a framework for structural multiscale geometric organization of graphs and subsets of R(n). We use diffusion semigroups to generate multiscale geometries in order to organize and represent complex structures. We show that appropriately selected eigenfunctions or scaling functions of Markov matrices, which describe local transitions, lead to macroscopic descriptions at different scales. The process of iterating or diffusing the Markov matrix is seen as a generalization of some aspects of the Newtonian paradigm, in which local infinitesimal transitions of a system lead to global macroscopic descriptions by integration. We provide a unified view of ideas from data analysis, machine learning, and numerical analysis.

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