Speaker
Description
The study of the interplay between statistics and geometry has expanded rapidly in both scope and influence over recent decades. The idea of manifold fitting can be traced back to the pioneering work of Hassler Whitney in the early 1930s. A fundamental question in this area is the following: given a data set, under what conditions can we construct a smooth (d)-dimensional surface (or manifold) that provides a good approximation to the data, and how can the quality of such a fit be quantified in terms of distance and smoothness?
In this talk, I will provide an overview of the manifold fitting problem, with particular emphasis on our recent work on principal nested submanifolds. This framework offers a flexible nonlinear approach to capturing the intrinsic geometric structure of high-dimensional data. I will also discuss several recent insights and developments at the intersection of geometry, statistics, and data science.