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Conformal and holomorphic barycenters in hyperbolic balls

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arxiv 2410.02257 v3 pith:S757TIT5 submitted 2024-10-03 math.DG

Vladimir Jacimovic , David Kalaj This is my paper · ORCID
classification math.DG
keywords barycentersballsbarycenterconformalholomorphichyperbolicmathbbnotions
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abstract

We introduce the notions of \textit{conformal barycenter} and \textit{holomorphic barycenter} of a measurable set $D$ in the hyperbolic ball. The two barycenters coincide in the disk, but they differ in multidimensional balls $\mathbb{C}^m \cong \mathbb{R}^{2m}$. These notions are counterparts of barycenters of measures on spheres, introduced by Douady and Earle in 1986.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Clustering in hyperbolic balls

    cs.LG 2025-01 reject novelty 4.0 of 10

    K-means and EM clustering for points in Poincaré hyperbolic balls are defined via conformal barycenters and Möbius distributions, with synthetic experiments in 2D and 3D.

  2. A group-theoretic framework for machine learning in hyperbolic spaces

    cs.LG 2025-01 reject novelty 4.0 of 10

    Introduces conformal and holomorphic barycenters and Möbius-type probability families on hyperbolic balls, together with hyperbolic gradient and maximum likelihood estimation algorithms.

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