Strong Convexity of Sets in Riemannian Manifolds

Damien Scieur, Thomas Kerdreux, Martínez-Rubio, Alexandre d'Aspremont, Sebastian Pokutta

Published 2023-12-06Version 1

Convex curvature properties are important in designing and analyzing convex optimization algorithms in the Hilbertian or Riemannian settings. In the case of the Hilbertian setting, strongly convex sets are well studied. Herein, we propose various definitions of strong convexity for uniquely geodesic sets in a Riemannian manifold. We study their relationship, propose tools to determine the geodesic strongly convex nature of sets, and analyze the convergence of optimization algorithms over those sets. In particular, we demonstrate that, when the Riemannian scaling inequalities hold, the Riemannian Frank-Wolfe algorithm enjoys a global linear convergence rate.