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

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.

authors

Coifman RR,Lafon S,Lee AB,Maggioni M,Nadler B,Warner F,Zucker SW

doi

10.1073/pnas.0500334102

keywords:

subject

Has Abstract

pub_date

2005-05-24 00:00:00

pages

7426-31

issue

21

eissn

0027-8424

issn

1091-6490

pii

0500334102

journal_volume

102

pub_type

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