On Approximating the Riemannian 1-Center

Abstract

In this paper, we generalize the simple Euclidean 1-center approximation algorithm of Badoiu and Clarkson (2003) to Riemannian geometries and study accordingly the convergence rate. We then show how to instantiate this generic algorithm to two particular cases: (1) hyperbolic geometry, and (2) Riemannian manifold of symmetric positive definite matrices.

23 pages

References in corpus (3)

• Riemannian Median and Its Estimation
• Hyperbolic Voronoi diagrams made easy
• Stochastic algorithms for computing means of probability measures

Cited by in corpus (12)

• On a Variational Definition for the Jensen-Shannon Symmetrization of Distances based on the Information Radius
• Mumford-Shah and Potts Regularization for Manifold-Valued Data with Applications to DTI and Q-Ball Imaging
• Clustering in Hilbert simplex geometry
• A numerical framework for elastic surface matching, comparison, and interpolation
• A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions
• Hilbert geometry of the Siegel disk: The Siegel-Klein disk model
• Interior-point methods on manifolds: theory and applications
• The hyperbolic Voronoi diagram in arbitrary dimension
• What is an inductive mean?
• Stochastic and incremental subgradient methods for convex optimization on Hadamard spaces
• Algorithms for distance problems in planar complexes of global nonpositive curvature
• Uncertainty Quantification of Set-Membership Estimation in Control and Perception: Revisiting the Minimum Enclosing Ellipsoid