{"ID":2923173,"CreatedAt":"2026-06-02T03:17:13.356150003Z","UpdatedAt":"2026-06-04T07:41:34.29888543Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.02116","arxiv_id":"2606.02116","title":"Retraction based regression methods on Riemannian manifolds","abstract":"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.","short_abstract":"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 retrac...","url_abs":"https://arxiv.org/abs/2606.02116","url_pdf":"https://arxiv.org/pdf/2606.02116v1","authors":"[\"Estefanía Loayza-Romero\",\"Benedikt Sibum\",\"Kathrin Welker\"]","published":"2026-06-01T11:48:09Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}