Gromov-Wasserstein Alignment: Statistics and Computation

Over the past decade, the statistical and computational properties of optimal transport (OT) have been systematically studied, driven, in part, to its broad applicability to data science. This program has culminated in an in depth understanding of the curse of dimensionality that OT distances suffer and spurred the development of computationally and statistically efficient proxies thereof via regularization. While OT distances enable a natural comparison between distributions on the same space, comparing datasets of different types (e.g., text and images) requires defining an ad hoc cost function which may not capture a meaningful correspondence between data points.

In this talk, I will survey the current statistical and computational landscape  for Gromov-Wasserstein (GW) distances, a framework which enables comparing abstract metric measure spaces based on their intrinsic metric structure and, as such, have seen widespread use in applications including comparing datasets of different types. I will present the first limit laws obtained for empirical GW distances, both with and without regularization, and describe consistent resampling schemes. Additionally, I will introduce the first algorithms for computing regularized GW distance subject to formal convergence guarantees. I will conclude by highlighting a number of open questions and future directions in the study of GW distances.

Joint work with Ziv Goldfeld and Kengo Kato.