Radial Compensation: Stable and Semantically Decoupled Generative Models on Riemannian Manifolds

Generative models on curved spaces rely on charts to map Euclidean spaces to manifolds. Exponential maps preserve geodesics but have stiff, radius-dependent Jacobians, while volume-preserving charts maintain densities but distort geodesic distances. Both approaches entangle curvature with model parameters, inflating gradient variance. In high-dimensional latent normalizing flows, the wrapped exponential prior can stretch radii far beyond the curvature scale, leading to poor test likelihoods and stiff solvers. We introduce Radial Compensation (RC), an information-geometric method that selects the base density in the tangent space so that the likelihood depends only on geodesic distance from a pole, decoupling parameter semantics from curvature. RC lets radial parameters retain their usual meaning in geodesic units, while the chart can be tuned as a numerical preconditioner. We extend RC to manifolds with known geodesic polar volume and show that RC is the only construction for geodesic-radial likelihoods with curvature-invariant Fisher information. We derive the Balanced-Exponential (bExp) chart family, balancing volume distortion and geodesic error. Under RC, all bExp settings preserve the same manifold density and Fisher information, with smaller dial values reducing gradient variance and flow cost. Empirically, RC yields stable generative models across densities, VAEs, flows on images and graphs, and protein models. RC improves likelihoods, restores clean geodesic radii, and prevents radius blow-ups in high-dimensional flows, making RC-bExp a robust default for likelihood-trained generative models on manifolds.