Strongly Convex Sets in Riemannian Manifolds
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.
Cite
Text
Scieur et al. "Strongly Convex Sets in Riemannian Manifolds." International Conference on Learning Representations, 2026.Markdown
[Scieur et al. "Strongly Convex Sets in Riemannian Manifolds." International Conference on Learning Representations, 2026.](https://mlanthology.org/iclr/2026/scieur2026iclr-strongly/)BibTeX
@inproceedings{scieur2026iclr-strongly,
title = {{Strongly Convex Sets in Riemannian Manifolds}},
author = {Scieur, Damien and Martínez-Rubio, David and Kerdreux, Thomas and d'Aspremont, Alexandre and Pokutta, Sebastian},
booktitle = {International Conference on Learning Representations},
year = {2026},
url = {https://mlanthology.org/iclr/2026/scieur2026iclr-strongly/}
}