Title: Function Estimation on Riemannian Manifolds
Abstract:
Natural signals such as speech or images are embedded
in a high dimensional space. Yet, there may be good reason to think that they
actually lie on a low dimensional manifold embedded in this space.
Consequently, clustering, classification, and pattern recognition ought to work
with functions that are invariantly and intrinsically defined on the manifold
rather than on the ambient space.
We take this point of view to develop a family of
algorithms for dimensionality reduction, clustering, and classification for the
case where data lies on an unknown Riemannian manifold embedded (isometrically) in a very high dimensional space. Using the Laplace
Beltrami operator and its eigen
functions, we will develop invariant maps for each of the above problems. We see that the approach provides a natural
framework for combining supervised and unsupervised learning. We present algorithms, convergence theorems,
and applications. Finally, we will appreciate how geometry and topology
generate invariant maps and statistics allow us to estimate the correct ones.