# Robust Estimation in metric spaces: Achieving Exponential Concentration with a Fr'echet Median

arXiv: https://arxiv.org/abs/2504.14161 · HTML: https://mlsanity.com/paper/2504.14161/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 78 references that are in the map.
- First posted on arXiv 19 Apr 2025.

## Builds on (most cited first, 9 of 9)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Sub-Gaussian estimators of the mean of a random vector](https://mlsanity.com/paper/1702.00482/) (2017)
- [Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tails](https://mlsanity.com/paper/2205.08494/) (2022)
- [A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions](https://mlsanity.com/paper/2302.08175/) (2023)
- [Robust and Scalable Bayes via a Median of Subset Posterior Measures](https://mlsanity.com/paper/1403.2660/) (2014)
- [Kernel Smoothing, Mean Shift, and Their Learning Theory with Directional Data](https://mlsanity.com/paper/2010.13523/) (2020)
- [Tukey's Depth for Object Data](https://mlsanity.com/paper/2109.00493/) (2021)
- [Generalized Median Graph via Iterative Alternate Minimizations](https://mlsanity.com/paper/1906.11009/) (2019)

## Built on by (most cited first, 1 of 1)

- [Metric space valued Fr'echet regression](https://mlsanity.com/paper/2602.05225/) (2026)

## Abstract (arXiv)

There is growing interest in developing statistical estimators that achieve exponential concentration around a population target even when the data distribution has heavier than exponential tails. More recent activity has focused on extending such ideas beyond Euclidean spaces to Hilbert spaces and Riemannian manifolds. In this work, we show that such exponential concentration in presence of heavy tails can be achieved over a broader class of parameter spaces called CAT(kappa) spaces, a very general metric space equipped with the minimal essential geometric structure for our purpose, while being sufficiently broad to encompass most typical examples encountered in statistics and machine learning. The key technique is to develop and exploit a general concentration bound for the Fr'echet median in CAT(kappa) spaces. We illustrate our theory through a number of examples, and provide empirical support through simulation studies.