Barycenters and a law of large numbers in Gromov hyperbolic spaces
Shin-ichi Ohta
Osaka University, Osaka, Japan; RIKEN Center for Advanced Intelligence Project (AIP), Tokyo, Japan
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Abstract
We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction property (the Wasserstein distance between probability measures provides an upper bound of the distance between their barycenters), a deterministic approximation of barycenters of uniform distributions on finite points, and a kind of law of large numbers. These generalize the corresponding results on CAT(0)-spaces, up to additional terms depending on the hyperbolicity constant.
Cite this article
Shin-ichi Ohta, Barycenters and a law of large numbers in Gromov hyperbolic spaces. Rev. Mat. Iberoam. 40 (2024), no. 3, pp. 1185–1206
DOI 10.4171/RMI/1483