Jon Niles-Weed is an assistant professor of Mathematics and Data Science at the Courant Institute of Mathematical Sciences and the Center for Data Science at NYU, where he is a core member of the Math and Data and STAT groups. He studies statistics, probability, and the mathematics of data science, with a focus on statistical and computational problems arising from data with geometric structure. Much of his recent work is dedicated to developing a statistical theory of optimal transport. Jon received his Ph.D. in Mathematics and Statistics from MIT, under the supervision of Philippe Rigollet. His research is supported in part by the National Science Foundation, Google Research, and an Alfred P. Sloan Foundation fellowship.

**Talk**: The Sketched Wasserstein Distance for mixture distributions

**Abstract**: The Sketched Wasserstein Distance is a new probability distance specifically tailored to finite mixture distributions, constructed by lifting any metric defined on the mixture components to a metric on the space of mixtures. This metric can be uniquely characterized as the most discriminative jointly convex metric compatible with the metric defined on the mixture components. We show that the resulting metric space is isomorphic to a classical Wasserstein space consisting of "sketches" of the mixture distribution. Leveraging a dual formulation, we prove a general bound on the estimation error of the Sketched Wasserstein Distance and obtain nearly minimax optimal estimation rates for the distance between discrete mixtures. In the context of discrete mixtures, we obtain explicit distributional limits for our estimators of the mixture weights and the Sketched Wasserstein Distance itself. We complement these theoretical results with a simulation study and data analysis, justifying the applicability of the Sketched Wasserstein Distance to statistical problems involving mixtures. Joint work with Bing and Bunea.