On some global topological aspects of manifold learning

Abstract

With the dual purpose of helping place in perspective the diverse approaches to manifold learning, and facilitating future research, this paper steps back and describes the manifold learning problem from a holistic perspective. It is argued that getting the homology right can be crucial to successful classification schemes based on the intrinsic geometry of the learnt manifold, and furthermore, a purely Bayesian approach will not be able to succeed at this in general. Simple examples are given to illustrate the inherent limitations of manifold learning.