A Riemannian Corollary of Helly's Theorem
Metric Geometry
2019-10-03 v2 Data Structures and Algorithms
Abstract
We introduce a notion of halfspace for Hadamard manifolds that is natural in the context of convex optimization. For this notion of halfspace, we generalize a classic result of Gr\"unbaum, which itself is a corollary of Helly's theorem. Namely, given a probability distribution on the manifold, there is a point for which all halfspaces based at this point have at least of the mass. As an application, the gradient oracle complexity of convex optimization is polynomial in the parameters defining the problem.
Keywords
Cite
@article{arxiv.1804.10738,
title = {A Riemannian Corollary of Helly's Theorem},
author = {Alexander Rusciano},
journal= {arXiv preprint arXiv:1804.10738},
year = {2019}
}