Embeddings of Riemannian manifolds with finite eigenvector fields of connection Laplacian

Published

Journal Article

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. We study the problem asking if one can embed manifolds into finite dimensional Euclidean spaces by taking finite number of eigenvector fields of the connection Laplacian. This problem is essential for the dimension reduction problem in manifold learning. In this paper, we provide a positive answer to the problem. Specifically, we use eigenvector fields to construct local coordinate charts with low distortion, and show that the distortion constants depend only on geometric properties of manifolds with metrics in the little Hölder space c2,α. Next, we use the coordinate charts to embed the entire manifold into a finite dimensional Euclidean space. The proof of the results relies on solving the elliptic system and providing estimates for eigenvector fields and the heat kernel and their gradients. We also provide approximation results for eigenvector field under the c2,α perturbation.

Full Text

Duke Authors

Cited Authors

  • Lin, CY; Wu, HT

Published Date

  • October 1, 2018

Published In

Volume / Issue

  • 57 / 5

International Standard Serial Number (ISSN)

  • 0944-2669

Digital Object Identifier (DOI)

  • 10.1007/s00526-018-1401-3

Citation Source

  • Scopus