A Riemannian Corollary of Helly's Theorem

04/28/2018

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by   Alexander Rusciano, et al.

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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 Grünbaum, 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 1/n+1 of the mass. As an application, the gradient oracle complexity of convex optimization is polynomial in the parameters defining the problem.