[PDE&A] Matthew Thorpe: Linear Approximation and Manifold Learning in Optimal Transport

04 April 2024 16:00 till 17:00 - Location: Snijderszaal (LB 01.010 EEMCS) | Add to my calendar

Optimal transport distances are popular due to their 'modelling assumptions'. But significant drawbacks such as a lack of off-the-shelf data analysis tools and high computation cost limit their use in practice. The idea behind linear optimal transport is to define an embedding from an optimal transport space to a Euclidean space that approximates the topology. We start the talk by reviewing this embedding for the Wasserstein distance. Bounds on linear approximation are not particularly good if the full Wasserstein space is considered. To better control the approximation we consider a submanifold of the Wasserstein space and show one can get local linearisation of the same order as one expects for Riemannian manifolds in Euclidean spaces. We finally consider linearisation in the Hellinger--Kantorovich space, an extension of the Wasserstein distance to unbalanced measures. This is joint work with Tianji Cai, Junyi Cheng, Keaton Hamm, Caroline Moosemueller and Bernhard Schmitzer.