Mathematics Colloquium
Nicolas Charon
Johns Hopkins University
Title: Geometric data analysis: a Riemannian perspective
Date: Friday, January 27, 2023
Place and Time: fsu.zoom.us/s/97976878227, 3:05-3:55 pm
Abstract.
Extending statistical analysis and machine learning methods to geometric data, such as datasets of curves or surfaces, is a challenging task due to the intricate structure of these objects and the specific set of group invariances that are involved. This talk will present two of the mainstream frameworks to endow spaces of submanifolds with Riemannian metrics: the intrinsic (elastic) model based on quotient Sobolev metrics and the extrinsic diffeomorphic approach derived from Grenander's shape space formalism. Despite their different characterisitics, both of these provide effective settings for the comparison and interpolation of shapes and, by extension, for the generalization to geometric data of concepts and methods such as atlas estimation, parallel transport, tangent PCA and LDA, clustering... The presentation will focus specifically on the variational and optimal control formulation of such problems, and notably the use of tools from geometric measure theory for that purpose, as well as the numerical aspects of the implementation on discrete shapes. I will also discuss some extensions of those models, in particular to tackle the issue of partial data observations. Lastly, the talk will conclude with a few perspectives on ongoing and future research directions.