A Riemannian Corollary of Helly's Theorem

TL;DR

This paper extends Helly's theorem to Hadamard manifolds by defining a natural halfspace concept, leading to new insights in convex optimization with polynomial gradient oracle complexity.

Contribution

It introduces a geometric generalization of Helly's theorem on Hadamard manifolds and applies it to establish polynomial bounds in convex optimization.

Findings

Existence of a point with significant halfspace mass in Hadamard manifolds

Generalization of Grünbaum's result to non-Euclidean settings

Polynomial gradient oracle complexity for convex optimization

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 $\frac{1}{n+1}$ of the mass. As an application, the gradient oracle complexity of convex optimization is polynomial in the parameters defining the problem.