Journal
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
Volume 105, Issue 6, Pages 1803-1808Publisher
NATL ACAD SCIENCES
DOI: 10.1073/pnas.0710175104
Keywords
spectral geometry; nonlinear dimensionality reduction
Categories
Funding
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [0802635] Funding Source: National Science Foundation
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We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g., with C-alpha metric). These coordinates are bi-Lipschitz on large neighborhoods of the domain or manifold, with constants controlling the distortion and the size of the neighborhoods that depend only on natural geometric properties of the domain or manifold. The proof of these results relies on novel estimates, from above and below, for the heat kernel and its gradient, as well as for the eigenfunctions of the Laplacian and their gradient, that hold in the non-smooth category, and are stable with respect to perturbations within this category. Finally, these coordinate systems are intrinsic and efficiently computable, and are of value in applications.
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