Retraction based regression methods on Riemannian manifolds
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910281564684288 |
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| author | Loayza-Romero, Estefanía Sibum, Benedikt Welker, Kathrin |
| author_facet | Loayza-Romero, Estefanía Sibum, Benedikt Welker, Kathrin |
| contents | Geodesic regression generalizes classical regression models to manifold-valued data by replacing affine models in Euclidean spaces with geodesic models on Riemannian manifolds. In this paper, we set up a framework for regression based on retractions instead of the Riemannian exponential map and its corresponding retraction-based distance. The associated optimization problem is posed on a subset of the tangent bundle which is why we additionally construct retractions on the tangent bundle induced by retractions on the underlying manifold. Our approach yields a more flexible formulation which is applicable beyond settings where the exponential map can be computed efficiently. As a proof of concept, we apply the developed framework to the (n-1)-dimensional p-norm sphere using the retraction by normalization to define the regression problem. The resulting optimization problem is solved using the Riemannian steepest descent method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02116 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Retraction based regression methods on Riemannian manifolds Loayza-Romero, Estefanía Sibum, Benedikt Welker, Kathrin Optimization and Control 53Z50, 49Q12, 65J22, 49M05, 65K05, 68T01 Geodesic regression generalizes classical regression models to manifold-valued data by replacing affine models in Euclidean spaces with geodesic models on Riemannian manifolds. In this paper, we set up a framework for regression based on retractions instead of the Riemannian exponential map and its corresponding retraction-based distance. The associated optimization problem is posed on a subset of the tangent bundle which is why we additionally construct retractions on the tangent bundle induced by retractions on the underlying manifold. Our approach yields a more flexible formulation which is applicable beyond settings where the exponential map can be computed efficiently. As a proof of concept, we apply the developed framework to the (n-1)-dimensional p-norm sphere using the retraction by normalization to define the regression problem. The resulting optimization problem is solved using the Riemannian steepest descent method. |
| title | Retraction based regression methods on Riemannian manifolds |
| topic | Optimization and Control 53Z50, 49Q12, 65J22, 49M05, 65K05, 68T01 |
| url | https://arxiv.org/abs/2606.02116 |