Stochastic Analysis Seminar – Gabriel Rioux (Imperial College London)

Over the past two decades, the statistical properties of optimal transport (OT) have been systematically studied, driven in part by OT’s broad applicability across data science and statistics. This body of work has identified the curse of dimensionality inherent in statistical estimation with OT distances and has led to the development of statistically efficient regularizations of OT. A recent line of work has provided a simple delta method-based framework from which limit laws for OT distances with and without regularization can be derived.

While OT distances enable a natural comparison between distributions on a common space, comparing datasets of different types — such as text and images — requires specifying an ad hoc cost function, which may fail to capture a meaningful correspondence between data points. To address this issue, Gromov-Wasserstein (GW) distances have been proposed, enabling a comparison of metric measure spaces based on their intrinsic structure. As a result, GW distances have found widespread use in applications involving heterogeneous data. Remarkably, limit laws for GW distances can be established using the same delta method-based framework.

This talk will survey these recent results under the unified framework along with statistical applications for the derived limit laws.

This is joint work with Ziv Goldfeld, Kengo Kato, and Ritwik Sadhu.