# On the distance between mean and geometric median in high dimensions

arXiv: https://arxiv.org/abs/2508.12926 · HTML: https://mlsanity.com/paper/2508.12926/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 14 references that are in the map.
- First posted on arXiv 18 Aug 2025.

## Builds on (most cited first, 3 of 3)

- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [Challenges of cellwise outliers](https://mlsanity.com/paper/2302.02156/) (2023)
- [Robust Score Matching](https://mlsanity.com/paper/2501.05105/) (2025)

## Abstract (arXiv)

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.