Vector Diffusion Maps and the Connection Laplacian.

Published

Journal Article

We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high-dimensional data sets, images, and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low-dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on a low-dimensional manifold ℳ (d) embedded in ℝ (p) , we prove the relation between VDM and the connection Laplacian operator for vector fields over the manifold.

Full Text

Duke Authors

Cited Authors

  • Singer, A; Wu, H-T

Published Date

  • August 2012

Published In

Volume / Issue

  • 65 / 8

PubMed ID

  • 24415793

Pubmed Central ID

  • 24415793

Electronic International Standard Serial Number (EISSN)

  • 1097-0312

International Standard Serial Number (ISSN)

  • 0010-3640

Digital Object Identifier (DOI)

  • 10.1002/cpa.21395

Language

  • eng