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Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps (2005)
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| Venue: | Proceedings of the National Academy of Sciences |
| Citations: | 257 - 45 self |
BibTeX
@INPROCEEDINGS{Coifman05geometricdiffusions,
author = {R. R. Coifman and S. Lafon and A. B. Lee and M. Maggioni and F. Warner and S. Zucker},
title = {Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps},
booktitle = {Proceedings of the National Academy of Sciences},
year = {2005},
pages = {7426--7431}
}
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Abstract
of contexts of data analysis, such as spectral graph theory, manifold learning, nonlinear principal components and kernel methods. We augment these approaches by showing that the diffusion distance is a key intrinsic geometric quantity linking spectral theory of the Markov process, Laplace operators, or kernels, to the corresponding geometry and density of the data. This opens the door to the application of methods from numerical analysis and signal processing to the analysis of functions and transformations of the data. Abstract. We provide a framework for structural multiscale geometric organization of graphs and subsets of Rn. 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. In Part I below, we provide a unified view of ideas from data analysis, machine learning and numerical analysis. In Part II [1], we augment this approach by introducing fast order-N algorithms for homogenization of heterogeneous structures as well as for data representation. 1.
Keyphrases
structure definition geometric diffusion harmonic analysis diffusion map data analysis markov matrix numerical analysis diffusion semigroups complex structure spectral theory structural multiscale geometric organization nonlinear principal component local transition system lead markov process different scale data representation global macroscopic description spectral graph theory heterogeneous structure local infinitesimal transition diffusion distance corresponding geometry order-n algorithm machine learning key intrinsic geometric quantity multiscale geometry part ii unified view laplace operator kernel method newtonian paradigm signal processing
