On the distance between mean and geometric median in high dimensions

TL;DR

This paper investigates the relationship between the mean and geometric median in high-dimensional spaces, showing they become close under certain dependence conditions, with theoretical bounds and empirical validation.

Contribution

It provides the first theoretical bounds on the distance between mean and geometric median in high dimensions, including a rate-matching expansion.

Findings

Distance between mean and median vanishes as dimension increases

Derived an upper bound that asymptotically approaches zero

Simulation results confirm theoretical predictions

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.