[Colloquium] [CS] Colloquia Mikhail Belkin/PhD Dissertation/7/21/03

[Donna Brooms]

~~~~~~~~This is a Reminder for today's~~~~~~~~~~
Dissertation Defense Talk
___________________________
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