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Embeddings of Riemannian manifolds with finite eigenvector fields of connection Laplacian

  • Chen-Yun Lin
  • Hau-Tieng Wu
Article
  • 80 Downloads

Abstract

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 \(c^{2,\alpha }\). 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 \(c^{2,\alpha }\) perturbation.

Mathematics Subject Classification

53C23 62-07 35K08 35J47 

Notes

Acknowledgements

Hau-tieng Wu’s research is partially supported by Sloan Research Fellow FR-2015-65363. Chen-Yun Lin would like to thank Thomas Nyberg for his helpful discussions. The authors acknowledge anonymous reviewers’ valuable comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Department of Mathematics and Department of Statistical ScienceDuke UniversityDurhamUSA

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