How Geometry Shapes Generative Model Evaluation on Riemannian Manifolds

Overview & Synopsis

On Riemannian manifolds, generative-model evaluation is geometry-dependent since curvature and dimension jointly determine distances and volume growth. Our framework systematically compares geodesic, ambient, and tangent-space versions of Wasserstein and MMD across different geometries. Using controlled-noise perturbations and Riemannian Flow Matching samples, we study metric agreement, noise sensitivity, and their relation to geometry-induced volume scaling. We find that no metric is uniformly reliable: evaluation distance can affect both measured discrepancies and noise sensitivity, varying across curvature and dimension. Our analysis leads to practical recommendations for reporting evaluation distances, comparing metric types, and calibrating metrics to the geometry of the samples.