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The Euclidean space notion of convex sets (and functions) generalizes toRiemannian manifolds in a natural sense and is called geodesic convexity.Extensively studied computational problems such as convex optimization andsampling in convex sets also have meaningful counterparts in the manifoldsetting. Geodesically convex optimization is a well-studied problem withongoing research and considerable recent interest in machine learning andtheoretical computer science. In this paper, we study sampling and convexoptimization problems over manifolds of non-negative curvature provingpolynomial running time in the dimension and other relevant parameters. Ouralgorithms assume a warm start. We first present a random walk based samplingalgorithm and then combine it with simulated annealing for solving convexoptimization problems. To our knowledge, these are the first algorithms in thegeneral setting of positively curved manifolds with provable polynomialguarantees under reasonable assumptions, and the first study of the connectionbetween sampling and optimization in this setting.

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