A Riemannian Corollary of Helly's Theorem

Rusciano, Alexander

doi:10.48550/arxiv.1804.10738

Cited by 1 publication

(1 citation statement)

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“…It is natural to ask if this technique can be extended to manifolds. [Rus18] proves a version of the Grünbaum inequality in the case of Hadamard manifolds, showing that there exist points such that any halfspaces through these points have large fraction of the volume of the set. Using this they show an oracle complexity result for optimization, where each oracle call returns a gradient at a center point.…”

Section: Introduction 1motivationmentioning

confidence: 93%

Sampling and Optimization on Convex Sets in Riemannian Manifolds of Non-Negative Curvature

Goyal,

Shetty

2019

Preprint

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The Euclidean space notion of convex sets (and functions) generalizes to Riemannian manifolds in a natural sense and is called geodesic convexity. Extensively studied computational problems such as convex optimization and sampling in convex sets also have meaningful counterparts in the manifold setting. Geodesically convex optimization is a well-studied problem with ongoing research and considerable recent interest in machine learning and theoretical computer science. In this paper, we study sampling and convex optimization problems over manifolds of non-negative curvature proving polynomial running time in the dimension and other relevant parameters. Our algorithms assume a warm start. We first present a random walk based sampling algorithm and then combine it with simulated annealing for solving convex optimization problems. To our knowledge, these are the first algorithms in the general setting of positively curved manifolds with provable polynomial guarantees under reasonable assumptions, and the first study of the connection between sampling and optimization in this setting.

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Section: Introduction 1motivationmentioning

confidence: 93%