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

TL;DR

This work introduces Radial Compensation (RC), an information-geometric framework that decouples curvature effects from radial semantics in generative models on curved manifolds. RC pre-warps the tangent-space base so that the manifold likelihood depends only on geodesic distance to a pole, yielding Fisher information that matches one-dimensional Euclidean radial models and enabling robust, curvature-invariant learning. The authors develop the Balanced-Exponential (bExp) chart family and the Geodesic-Corrected Lambert (GCL) chart, which balance volume distortion and geodesic fidelity while preserving the radial semantics under RC; these charts function as numerical preconditioners that reduce gradient variance and CNF stiffness without changing the underlying density. They prove RC’s uniqueness within isotropic scalar–Jacobian bases, establish a balanced-polar pushforward for general manifolds with known polar volume, and demonstrate practical benefits across synthetic spheres, hyperspherical VAEs, mixed-curvature VAEs, latent CNFs on images, hyperbolic graph models, and SE(3)-style protein models. Empirically, RC improves likelihoods and stabilizes training across densities, restores clean geodesic radii, and prevents radius blow-ups, making RC–bExp a robust default for likelihood-trained generative models on manifolds and pointing to a general toolkit for geometry-aware learning.

Abstract

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.

Figures (13)

• Figure 1: Conceptual RC pipeline: a 1D radial base $\varphi_\theta$ on $\mathbb{R}_+$ is radially compensated so that geodesic radii $R=d(p,q)$ preserve the target semantics, then mapped to the manifold through an azimuthal chart $T$ (Exp / bExp$_\alpha$ / GCL), where the dial $\alpha$ tunes numerical conditioning without changing the geodesic–radial law.
• Figure 2: Exp (raw) vs. RC--Exp for a Gaussian base in the tangent. Top left: samples from a $2$-D isotropic Gaussian in the tangent plane $T_pM$. Both constructions start from the same Euclidean base; the labels Exp (raw) and RC--Exp indicate only which chart is used downstream. Top right: pushforward of these samples to the sphere $\mathbb{S}^2$ via the exponential map. The standard wrapped Gaussian obtained with Exp (raw) concentrates mass closer to the pole, shrinking the intended cap and pulling the geodesic radius $R=d(p,q)$ to smaller mean and variance, whereas RC--Exp yields a thicker cap at the prescribed geodesic scale. Bottom left: the same experiment on hyperbolic space $\mathbb{H}^2$, visualised in the Poincaré disk. Here Exp (raw) pushes mass further out, producing heavier tails and larger mean radius, while RC--Exp concentrates samples near the target scale. Bottom right: histograms of $R=d(p,q)$ on $\mathbb{S}^2$ and $\mathbb{H}^2$. Under RC--Exp the empirical law of $R$ matches the chosen one–dimensional radial target (same mean and variance in geodesic units), whereas Exp (raw) exhibits curvature–induced distortions due to the polar factor $(s_\kappa(R)/R)^{n-1}$. In particular, a parameter $\sigma$ that is intended to control geodesic spread implicitly depends on curvature, dimension, and chart, undermining hyperparameter transfer across geometries.
• Figure 3: Conceptual comparison of standard wrapped priors, ad-hoc fixes, and Radial Compensation (RC). Within the isotropic spherically symmetric (scalar--Jacobian) class, only RC simultaneously yields geodesic–radial models and chart–invariant Fisher information for the radial parameters (Theorem \ref{['thm:rc-uniqueness']}).
• Figure 4: Impossibility triangle. Within isotropic scalar–Jacobian models, any chart/base pair can satisfy at most two of: (i) geodesic–radial likelihoods, (ii) chart–invariant Fisher information in radial parameters, (iii) isotropy in the tangent space. RC is the unique way to realise all three simultaneously.
• Figure 5: Radial Compensation as a pre-warped prior.Left: in the tangent space, RC (orange) multiplies the target radial law $\phi(r)$ by the chart Jacobian $J_T(r)$, pushing mass outward and thickening the tails relative to the naive Euclidean base (blue dashed). Right: on hyperbolic space $H^{n}$, the naive Exp (raw) pushforward (orange dashed) is pulled toward the pole, with geodesic mean $\approx 1.47$ instead of the intended $\approx 2.0$. RC cancels the hyperbolic volume factor, so its pushforward (green) coincides with the target manifold radial law $\phi(R)$ (dotted), preserving parameter semantics (mean and scale) in geodesic units. RC takes the complexity hit in the latent space so that the manifold distribution remains simple and statistically meaningful.

Theorems & Definitions (75)

• Theorem 1: RC invariances: likelihood, Fisher, and KL
• Remark 1: Scope
• Theorem 2: Chart–invariance within the scalar–Jacobian class
• Theorem 3: Balanced polar pushforward on general manifolds
• Theorem 4: Characterization of RC
• Corollary 1: RC uniquely realises chart- and curvature-invariant radial semantics
• proof
• Corollary 2: RC is necessary and sufficient for Fisher-efficient radial learning
• Theorem 5: Implicit RC from likelihood matching
• Proposition 1: Representation of scalar--Jacobian azimuthal charts