Approximation of Riemannian Distances and Applications to Distance-Based Learning on Manifolds

Several important algorithms for machine learning and data analysis use pairwise distances as input. On Riemannian manifolds these distances may be prohibitively costly to compute, in particular for large datasets. To tackle this problem, we propose a distance approximation which requires only a linear number of geodesic boundary value problems to be solved. The approximation is constructed by fitting a two-dimensional model space with constant curvature to each pair of samples. We demonstrate the usefulness of our approach in the context of shape analysis on landmarks spaces.