Geometry and barycenters in large CAT(κ) spaces
Abstract
We study the geometry and resulting properties of Fréchet functions and barycenters in metric spaces with curvature bounded above by a positive constant in the sense of Alexandrov. Existing results typically rely on the assumption that the support is small enough that the squared distance, and thus the Fréchet function, is strongly geodesically convex. We work in the sharper regime where existence and uniqueness of the barycenter are guaranteed, but where convexity typically fails. We develop new geometric properties of these spaces, including a quantitative four-point inequality and a geodesic stability result. Using these tools, we establish new finite sample error bounds for two statistical estimators: the empirical barycenter and the iterated barycenter, as well as new results on sparse representations and approximations of barycenters.
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BibTeX
@misc{boukhateb2026geometry,
title = {{Geometry and barycenters in large $\CAT(κ)$ spaces}},
author = {Yassine Boukhateb and Victor-Emmanuel Brunel and Austin J. Stromme},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17909},
eprint = {2610.17909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17909},
note = {CC0},
}
RIS
TY - UNPB
TI - Geometry and barycenters in large CAT(κ) spaces
AU - Yassine Boukhateb
AU - Victor-Emmanuel Brunel
AU - Austin J. Stromme
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17909
UR - https://arxiv.org/abs/2610.17909
AB - We study the geometry and resulting properties of Fréchet functions and barycenters in metric spaces with curvature bounded above by a positive constant in the sense of Alexandrov. Existing results typically rely on the assumption that the support is small enough that the squared distance, and thus the Fréchet function, is strongly geodesically convex. We work in the sharper regime where existence and uniqueness of the barycenter are guaranteed, but where convexity typically fails. We develop new geometric properties of these spaces, including a quantitative four-point inequality and a geodesic stability result. Using these tools, we establish new finite sample error bounds for two statistical estimators: the empirical barycenter and the iterated barycenter, as well as new results on sparse representations and approximations of barycenters.
ER -