Journal
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
Volume 102, Issue 21, Pages 7426-7431Publisher
NATL ACAD SCIENCES
DOI: 10.1073/pnas.0500334102
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- NINDS NIH HHS [R01 NS100049] Funding Source: Medline
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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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