Radial Compensation: The Inverse Base-Distribution Problem for Chart-Based Generative Models on Riemannian Manifolds

The paper addresses a flaw in latent‑variable generative models on Riemannian manifolds such as spheres and hyperbolic spaces, where the usual practice of sampling a Gaussian in a tangent space and mapping it onto the manifold inadvertently imposes a fixed chi‑distribution on distances from a base point. The authors formulate and solve the inverse problem: given a desired distance distribution, they derive the exact tangent‑space density that yields it, prove its uniqueness for isotropic, chart‑independent likelihoods, and provide a lower bound on the cost of ignoring this issue in variational autoencoders. Experiments with exact normalization audits show that the compensated prior is chart‑invariant, stable across scales, and leads to significant improvements in curvature recovery and protein‑orientation likelihoods. whyItMatters":"By correcting the implicit distance distribution, the method enables more accurate and stable generative modeling on curved spaces, directly improving performance on tasks such as protein orientation."