Orientability and diffusion maps

Amit Singer, Hau Tieng Wu

Research output: Contribution to journalArticle

21 Scopus citations

Abstract

One of the main objectives in the analysis of a high dimensional large data set is to learn its geometric and topological structure. Even though the data itself is parameterized as a point cloud in a high dimensional ambient space Rp, the correlation between parameters often suggests the "manifold assumption" that the data points are distributed on (or near) a low dimensional Riemannian manifold Md embedded in Rp, with d≪p. We introduce an algorithm that determines the orientability of the intrinsic manifold given a sufficiently large number of sampled data points. If the manifold is orientable, then our algorithm also provides an alternative procedure for computing the eigenfunctions of the Laplacian that are important in the diffusion map framework for reducing the dimensionality of the data. If the manifold is non-orientable, then we provide a modified diffusion mapping of its orientable double covering.

Original languageEnglish (US)
Pages (from-to)44-58
Number of pages15
JournalApplied and Computational Harmonic Analysis
Volume31
Issue number1
DOIs
StatePublished - Jul 2011

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

Keywords

  • Diffusion maps
  • Dimensionality reduction
  • Orientability

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