Radial Compensation: Fixing Radius Distortion in Chart-Based Generative Models on Riemannian Manifolds

TL;DR

This paper introduces Radial Compensation (RC), a method for fixing radius distortion in chart-based generative models on Riemannian manifolds, enhancing stability and interpretability.

Contribution

It proposes a novel RC approach that decouples statistical modeling from numerical conditioning in manifold generative models.

Findings

RC improves numerical stability in manifold VAEs and flows.

RC enables explicit control over geodesic radius distribution.

Balanced exponential charts enhance conditioning without altering manifold density.

Abstract

We study the base distribution in chart-based generative models on Riemannian manifolds. Standard methods sample in Euclidean tangent space and then map the sample to the manifold with a chart. This is convenient, but it changes the meaning of distance: the same tangent-space scale can correspond to different geodesic radii, i.e. shortest-path distances from a reference point on the manifold, under different charts, curvatures, and dimensions. Within isotropic, scalar-Jacobian azimuthal charts, we show that no base distribution can simultaneously preserve geodesic-radial likelihoods, chart-invariant radial Fisher information, and tangent-space isotropy unless it has a specific form, which we call Radial Compensation (RC). RC chooses the tangent-space base so that the model realizes a user-specified one-dimensional law for the geodesic radius, and leaves the chart available as a numerical preconditioner. This gives more stable training and cleaner curvature estimates, because curvature no longer has to compensate for distortions introduced by the chart. We also introduce balanced exponential charts, which improve conditioning without changing the realized manifold density under RC. This decouples the statistical meaning of the model, the law of the geodesic radius, from its numerical conditioning, which is governed by the chart Jacobian: chart choice becomes a numerical preconditioner rather than a hidden modeling decision. Across manifold variational autoencoders and continuous normalizing flows, RC matches the intended radius behavior, improves numerical stability, and makes learned curvature easier to interpret.