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.