Abstract

We provide a framework for structural multiscale geometric organization of graphs and subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathbb{R}}^{n}\end{equation*}\end{document} . 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.

Keywords

GeneralizationEigenfunctionStatistical physicsMarkov chainComputer scienceDiffusionInfinitesimalDiffusion mapMatrix (chemical analysis)Markov processScalingTheoretical computer scienceMathematicsAlgorithmEigenvalues and eigenvectorsArtificial intelligenceGeometryPhysicsMathematical analysisMachine learning

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Publication Info

Year
2005
Type
article
Volume
102
Issue
21
Pages
7426-7431
Citations
1704
Access
Closed

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Cite This

Ronald R. Coifman, Stéphane Lafon, A. B. Lee et al. (2005). Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps. Proceedings of the National Academy of Sciences , 102 (21) , 7426-7431. https://doi.org/10.1073/pnas.0500334102

Identifiers

DOI
10.1073/pnas.0500334102
PMID
15899970
PMCID
PMC1140422

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Data completeness: 81%