期刊
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
卷 102, 期 21, 页码 7426-7431出版社
NATL ACAD SCIENCES
DOI: 10.1073/pnas.0500334102
关键词
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资金
- NINDS NIH HHS [R01 NS100049] Funding Source: Medline
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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