CSAIL Event Calendar: Previous Series
Geometry, Perception and Learning
Speaker: Partha Niyogi , University of Chicago
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 eigenfunctions, 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.