Fréchet Means in Infinite Dimensions

Jaffe Adam, UC Berkeley, Department of Statistics

The Fréchet mean (also called the barycenter or the center of mass) is a fundamental notion of central tendency for data living in an abstract metric space, and, as such it has received much attention as an estimator of interest in non-Euclidean statistics. However, the emerging probabilistic theory of Fréchet means is mostly limited to the "finite-dimensional" setting, save for a few isolated results in particular "infinite-dimensional" settings of interest. Presently, we introduce a geometric condition for a general infinite-dimensional setting under which one can prove a continuity result that implies a strong law of large numbers, an ergodic theorem, a large deviations principle, and more. We also reduce the moment assumptions from the existing literature to the provably minimal possible. We conclude by proving novel results or strengthening existing results in three applications: metric projections in approximation theory, Wasserstein barycenters in optimal transport, and image processing in pattern theory.

Area: CS47 - Statistical inference in infinite-dimensional spaces (Alessia Caponera)

Keywords: Fréchet mean, non-Euclidean statistics, optimal transport

Please Login in order to download this file