Robust, sub-Gaussian mean estimators in metric spaces by Prof Roberto Imbuzeiro Oliveira

Abstract

Estimating the mean of a random vector from i.i.d. data has received considerable attention. When the data take values in more general metric spaces, an appropriate extension of the notion of the mean is the Fréchet mean. While asymptotic properties of the most natural Fréchet mean estimator (the empirical Fréchet mean) have been
thoroughly researched, non-asymptotic performance bounds have only been studied recently.

This talk considers the performance of estimators of the Fréchet mean in general metric spaces under possibly heavy-tailed and contaminated data. In such cases, the empirical Fréchet mean is a poor estimator. We propose a general estimator based on high-dimensional extensions of trimmed means and prove general performance bounds. Unlike all previously established bounds, ours generalize the optimal bounds known for Euclidean data. Much like in the Euclidean case, the optimal accuracy is governed by two “variance” terms: a “global variance” term that is independent of the prescribed confidence, and a potentially much smaller, confidence-dependent “local variance” term. We apply our results for metric spaces with curvature bounded from below, such as Wasserstein spaces, and for uniformly convex Banach spaces.

The talk is based on arXiv:2509.13606, which is joint work with Daniel Bartl (NUS), Gabor Lugosi (ICREA/UPF), and Zoraida Rico (Bocconi).

Biography

Roberto Imbuzeiro Oliveira is a "pesquisador titular" (equivalent to full professor) and a member of the Data Science group at the Instituto de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro. His interests span Probability theory and its connections with Statistics, Discrete Mathematics, Optimization and other topics. In 2025, Roberto was named a Fellow of the Institute of Mathematical Statistics (IMS) "for being a leading researcher in probability and mathematical statistics with a rarely seen breadth of interests and expertise." 

Roberto has served the community in various capacities, including as the editor-in-chief of ALEA; an associate editor of Bernoulli and Operations Research; a senior PC member of the Conference on Learning Theory (COLT); and a member of the Board of Directors of the Brazilian Mathematical Society (SBM).