Strongly Convex Sets in Riemannian Manifolds

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

Keywords: Strong convexity, Riemannian, Manifold, Frank-Wolfe, optimization, nonconvex

TL;DR: We propose various definitions of strong convexity for uniquely geodesic sets in a Riemannian manifold, examine their relationships, introduce tools to identify strong convexity, and analyze the convergence of optimization algorithms over these sets

Abstract: Strong convexity plays a key role in designing and analyzing convex optimization algorithms and is well-understood in Hilbert spaces. However, the notion of strongly convex sets beyond Hilbert spaces remains unclear. In this paper, we propose various definitions of strong convexity for uniquely geodesic sets in a Riemannian manifold, examine their relationships, introduce tools to identify geodesically strongly convex sets, and analyze the convergence of optimization algorithms over these sets. In particular, we show that the Riemannian Frank-Wolfe algorithm converges linearly when the Riemannian scaling inequalities hold.

Primary Area: optimization

Submission Number: 12599