Barycenters and a law of large numbers in Gromov hyperbolic spaces

• Shin-ichi Ohta ^[1]
  1. [1] Osaka University

     Osaka University

     Kita Ku, Japón

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 40, Nº 3, 2024, págs. 1185-1206
• Idioma: inglés
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• Resumen

  • 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.