# Finite sample bounds for barycenter estimation in geodesic spaces

arXiv: https://arxiv.org/abs/2502.14069 · HTML: https://mlsanity.com/paper/2502.14069/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 56 references that are in the map.
- First posted on arXiv 19 Feb 2025.

## Builds on (most cited first, 8 of 8)

- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [Wasserstein barycenters are NP-hard to compute](https://mlsanity.com/paper/2101.01100/) (2021)
- [Wasserstein barycenters can be computed in polynomial time in fixed dimension](https://mlsanity.com/paper/2006.08012/) (2020)
- [Differential Privacy Over Riemannian Manifolds](https://mlsanity.com/paper/2111.02516/) (2021)
- [Towards Riemannian Accelerated Gradient Methods](https://mlsanity.com/paper/1806.02812/) (2018)
- [Minimizing Dynamic Regret on Geodesic Metric Spaces](https://mlsanity.com/paper/2302.08652/) (2023)

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

- [K-means with learned metrics](https://mlsanity.com/paper/2603.14601/) (2026)

## Abstract (arXiv)

We study the problem of estimating the barycenter of a distribution given i.i.d. data in a geodesic space. Assuming an upper curvature bound in Alexandrov's sense and a support condition ensuring the strong geodesic convexity of the barycenter problem, we establish finite-sample error bounds in expectation and with high probability. Our results generalize Hoeffding- and Bernstein-type concentration inequalities from Euclidean to geodesic spaces. Building on these concentration inequalities, we derive statistical guarantees for two efficient algorithms for the computation of barycenters.