This thesis consists of two parts, in both of which eigenfunctions and eigenvalues of the Laplace operator on Riemannian manifolds play an important role. The first part of the thesis studies embeddings of manifolds into Euclidean space by heat kernels and eigenfunctions of the Laplacian, whereas the second part analyzes the behavior of eigenvalues as manifolds converge to a limit space.
To understand structure or the most important features in data gathered in large experiments or for the purpose of machine learning, classically linear methods such as Principal Component Analysis are used. However, when the data satisfies nonlinear constraints, or lies on a manifold, nonlinear methods are required to pick up the relevant structure. Several such algorithms in nonlinear data analysis use the eigenfunctions of the Laplace operator on the data graph to embed the data in a lower-dimensional Euclidean space. The first part of the thesis is devoted to developing an understanding of the continuum versions of these algorithms. In particular, we bound the complexity of such algorithms in terms of geometric information. That is, we show that the number of eigenfunctions or heat kernels needed can be bounded in terms of the dimension, the volume, the injectivity radius, a lower bound on the Ricci curvature, and a tolerance for the dilatation.
In the second part of the thesis, we study the behavior of the spectrum of the Laplace operator on (generalizations of) Riemannian manifolds, as these manifolds converge to a limit space. Previous results by Fukaya and by Cheeger and Colding show that under certain bounds on the curvature of the manifolds, the eigenvalues of the Laplace operator are continuous with respect to measured Gromov-Hausdorff convergence. We are interested in the behavior of the spectrum under flat and intrinsic flat convergence. First, we show the semicontinuity of eigenvalues of the Laplace operator on Riemannian manifolds under flat convergence in Euclidean space, if the volume is conserved. In the more general case of intrinsic flat convergence, we can find analogous results for min-max values that arise from a variational problem involving a Dirichlet energy. In the case of closed Riemannian manifolds, these min-max values correspond to the eigenvalues of the Laplace operator.
|Commitee:||Cheeger, Jeff, Deift, Percy A., Kleiner, Bruce A., Kohn, Robert V.|
|School:||New York University|
|School Location:||United States -- New York|
|Source:||DAI-B 76/01(E), Dissertation Abstracts International|
|Keywords:||Eigenfunctions, Laplace operators, Nonlinear data analysis, Riemannian manifolds, Spectral geometry|
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