Approximated logarithmic maps on Riemannian manifolds and their applications

Recently, optimization problems on Riemannian manifolds involving geodesic distances have been attracting considerable research interest. To compute geodesic distances and their Riemannian gradients, we can use logarithmic maps. However, the computational cost of logarithmic maps on Riemannian manifolds is generally higher than that on the Euclidean space. To overcome this computational issue, we propose approximated logarithmic maps. We prove that the definition is closely related to the inverse retractions. Numerical experiments for computing the Riemannian center of mass show that the proposed approximation significantly reduces the computational time while maintaining appropriate precision if the data diameter is sufficiently small.