[Colloquium] [CS] Colloquia Mikhail Belkin/PhD Dissertation/7/21/03
Donna Brooms
donna at cs.uchicago.edu
Fri Jul 18 14:35:58 CDT 2003
Dissertation Defense Announcement
___________________________
Ph.D. Candidate: Mikhail Belkin
Department of Mathematics
Date: Monday, July 21, 2003
Time: 3:00 p.m.
Place: Ryerson 251
Title: Problems of Learning on Manifolds
Abstract:
This thesis discusses the general problem of learning a function on a
manifold given by data points. Such problems arise in clustering,
classifying, and generally analyzing high dimensional data sets.
The space of functions on a Riemannian manifold has a family of
smoothness functionals and a canonical basis
associated to the Laplace-Beltrami operator. Moreover, the Laplace-
Beltrami operator can be reconstructed with certain convergence
guarantees when the manifold is only known through the sampled data
points. This allows the techniques of regularization and Fourier
analysis to be applied to functions defined on data. A convergence
result is proved for the case when data is sampled from a compact
submanifold of $\R^k$. Several practical applications are considered.
Advisor: Partha Niyogi
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