Seminar Title: Sliced Optimal Transport and Statistics
Abstract
Sliced optimal transport (SOT) provides a statistically and computationally scalable framework for comparing and transforming distributions. In this talk, I will motivate the use of SOT in statistics through two problems. The first problem concerns multiple multivariate density–density regression, where both predictors and responses are multivariate distributions. I will introduce a Bayesian inference model with a generalized likelihood based on the sliced Wasserstein (SW) distances between observed responses and fitted distributions. Each fitted distribution is constructed as a SW barycenter of push-forwards of the corresponding predictors. I will discuss related properties of this model and illustrate how it can reveal patterns of cell–cell communication from single-cell data. The second problem is about summarizing posterior inference on random partitions in nonparametric Bayesian mixture models. The objective is to summarize posterior uncertainty by identifying a representative partition. I introduce a novel approach that addresses the label-switching problem by deriving a point estimate for the random mixing measure, which implies both the partition and the mixture density. The point estimate for the mixing measure is based on a novel geometrically meaningful projection for product manifolds, leading to two new SOT metrics for Gaussian mixing measures. I will conclude by discussing some future directions.