Strong Convexity of Sets in Riemannian Manifolds
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910744081072128 |
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| author | Scieur, Damien Martínez-Rubio, David Kerdreux, Thomas d'Aspremont, Alexandre Pokutta, Sebastian |
| author_facet | Scieur, Damien Martínez-Rubio, David Kerdreux, Thomas d'Aspremont, Alexandre Pokutta, Sebastian |
| contents | Curvature properties of convex objects, such as strong convexity, 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 the Riemannian Frank-Wolfe algorithm enjoys a global linear convergence rate when the Riemannian scaling inequalities hold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_03583 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Strong Convexity of Sets in Riemannian Manifolds Scieur, Damien Martínez-Rubio, David Kerdreux, Thomas d'Aspremont, Alexandre Pokutta, Sebastian Optimization and Control Curvature properties of convex objects, such as strong convexity, 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 the Riemannian Frank-Wolfe algorithm enjoys a global linear convergence rate when the Riemannian scaling inequalities hold. |
| title | Strong Convexity of Sets in Riemannian Manifolds |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2312.03583 |