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

TL;DR

This paper develops the first polynomial-time algorithms for sampling and convex optimization on Riemannian manifolds with non-negative curvature, connecting sampling techniques with optimization in this geometric setting.

Contribution

It introduces novel algorithms for sampling and optimization on positively curved manifolds, establishing polynomial guarantees and linking these two fundamental problems.

Findings

First algorithms with provable guarantees for sampling in non-negatively curved manifolds

Combined sampling and simulated annealing approach for convex optimization

Polynomial running time in dimension and relevant parameters

Abstract

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.