A Geometric Perspective on Machine Learning

Natural data (speech signals, images, etc.) live in very
high dimensional spaces. However, there has always been the strong
intuition that there are only a few explanatory degrees of freedom.
One way to formalize this intuition is to model the data as lying
on or near a low dimensional manifold embedded in the original high
dimensional space. This point of view gives rise to a new class
of geometrically motivated learning algorithms known as manifold
learning algorithms.

I will discuss a framework for manifold learning
and show how the classical problems of statistical machine learning
such as dimensionality reduction, clustering, classification, and
regression may be treated within this framework. We will see how
various geometric and topological invariants of an underlying
unknown manifold may be estimated from random samples, and how
ideas from geometry, statistics, and computer science may come
together in a new and interesting way in this setting.