Titles and abstracts

Abstract

Abstract

Abstract

Abstract

In this talk, we revisit the classical theory of variance bounds for unbiased estimators and develop an analogous framework for a different notion of estimator stability, which we term sensitivity. Roughly speaking, the sensitivity of an estimator quantifies how much its output changes when its inputs are perturbed slightly. In the same way that classical inequalities like the Cramer-Rao lower bound for the variance of unbiased estimators can be interpreted geometrically via the Fisher-Rao geometry, our notion of sensitivity admits analogous inequalities and theory, only that now they arise from Wasserstein geometry. We will discuss this Wasserstein-Cramer-Rao lower bound and its associated notion of Wasserstein Fisher information (which are closely related to notions introduced by Li and Zhao, 2023), introduce the notion of Wasserstein "exponential" families and their defining properties, and introduce the concept of Wasserstein "MLEs". In particular, we will discuss that the Wasserstein MLE is, generically, asymptotically efficient for sensitivity, in the sense that it achieves the corresponding Wasserstein-Cramer-Rao lower bound. Our theory reveals new optimality properties for existing estimators and, in other cases, reveals entirely new estimators.  

This is joint work with Adam Quinn Jaffe and Bodhisattva Sen.

Abstract

Abstract

Abstract

Abstract

Abstract

Transportation of measure underlies many contemporary methods in machine learning and statistics. Sampling, which is a fundamental building block in computational science, can be done efficiently given an appropriate measure-transport map. We ask: what is the effect of using approximate maps in such algorithms? At the core of our analysis is the theory of optimal transport regularity, approximation theory, and an emerging class of inequalities, previously studied in the context of uncertainty quantification (UQ).

Abstract