DMCA
Diffusion Maps, Spectral Clustering and Reaction (2006)
| Venue: | Applied and Computational Harmonic Analysis: Special issue on Diffusion Maps and Wavelets |
| Citations: | 90 - 14 self |
BibTeX
@INPROCEEDINGS{Nadler06diffusionmaps,,
author = {Boaz Nadler and Stephane Lafon and Ronald R. Coifman and Ioannis G. Kevrekidis},
title = {Diffusion Maps, Spectral Clustering and Reaction},
booktitle = {Applied and Computational Harmonic Analysis: Special issue on Diffusion Maps and Wavelets},
year = {2006},
pages = {2006}
}
Share
 |
 |
 |
 |
||
OpenURL
Abstract
A central problem in data analysis is the low dimensional representation of high dimensional data, and the concise description of its underlying geometry and density. In the analysis of large scale simulations of complex dynamical systems, where the notion of time evolution comes into play, important problems are the identification of slow variables and dynamically meaningful reaction coordinates that capture the long time evolution of the system. In this paper we provide a unifying view of these apparently different tasks, by considering a family of di#usion maps, defined as the embedding of complex (high dimensional) data onto a low dimensional Euclidian space, via the eigenvectors of suitably defined random walks defined on the given datasets. Assuming that the data is randomly sampled from an underlying general probability distribution p(x) = e -U(x) , we show that as the number of samples goes to infinity, the eigenvectors of each di#usion map converge to the eigenfunctions of a corresponding di#erential operator defined on the support of the probability distribution. Di#erent normalizations of the Markov chain on the graph lead to di#erent limiting di#erential operators.
Keyphrases
spectral clustering diffusion map graph lead di usion map converge di erential operator random walk large scale simulation low dimensional euclidian space unifying view low dimensional representation di erent normalization long time evolution meaningful reaction time evolution concise description general probability distribution high dimensional data corresponding di erential operator slow variable di usion map probability distribution markov chain underlying geometry complex dynamical system high dimensional important problem different task central problem data analysis
