Work

On the distance between mean and geometric median in high dimensions

Year: 2026

Type: article

Abstract: The geometric median, a notion of “center” for multivariate distributions, has gained recent attention in robust statistics and machine learning. Although conceptually distinct from the mean (i.e., expectation), we demonstrate that both are very close in high dimensions when the dependence between the distribution components is suitably controlled. Concretely, we find an upper bound on the distance that vanishes with the dimension asymptotically, and derive a rate-matching first order expansion of the geometric median components. Simulations illustrate and confirm our results.

Source: Statistics & Probability Letters

DOI: 10.1016/j.spl.2026.110679

Authors: Richard Schwank, Mathias Drton

Institutions: Technical University of Munich, Munich Center for Machine Learning

Language: English

Topic: Advanced Statistical Methods and Models

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,180

Funders: Deutscher Akademischer Austauschdienst, Bundesministerium für Bildung und Forschung, Bayerisches Staatsministerium für Bildung und Kultus, Wissenschaft und Kunst, Technische Universität München