aiqbench analysis
This research addresses a critical flaw in latent-variable models operating on non-Euclidean manifolds, where standard Gaussian priors inadvertently enforce a fixed chi distribution rather than the intended distance metric. By deriving a closed-form solution for the correct tangent density, the authors enable chart-invariant likelihoods and significantly improve performance in tasks like protein orientation modeling. This advancement is essential for building robust generative models that accurately represent hierarchical data on spheres and hyperbolic spaces without collapsing to boundary artifacts.
Radial Compensation: The Inverse Base-Distribution Problem for Chart-Based Generative Models on Riemannian Manifolds
arXiv:2511.14056v3 Announce Type: replace-cross Abstract: Latent-variable models on spheres and hyperbolic spaces usually draw a Gaussian in the tangent space at a base point and push it onto the manifold. On these spaces the distance from the base point is the coordinate that carries meaning: depth in a hierarchy, the angle of a rotation, the deviation of a protein frame from a reference. We show that the standard construction silently replaces whatever distance distribution the modeler intended with a fixed one, a scaled chi law whose shape no setting of the scale can change. We then solve the reverse problem. Given the intended distance distribution, we derive in closed form the tangent density that realizes it, prove it is the only isotropic choice with chart-independent likelihoods for a broad class of charts, and prove a lower bound with explicit constants on what ignoring the problem costs a variational autoencoder. Experiments backed by an exact per-run normalization audit confirm that the compensated prior is invariant to the chart and stable across scales, every wrapped baseline we train collapses to the boundary of its chart, curvature becomes recoverable where the wrapped prior fails and protein-orientation likelihood improves from 2.58 to 0.87 nats at identical accuracy.
Abstract:Latent-variable models on spheres and hyperbolic spaces usually draw a Gaussian in the tangent space at a base point and push it onto the manifold. On these spaces the distance from the base point is the coordinate that carries meaning: depth in a hierarchy, the angle of a rotation, the deviation of a protein frame from a reference. We show that the standard construction silently replaces whatever distance distribution the modeler intended with a fixed one, a scaled chi law whose shape no setting of the scale can change. We then solve the reverse problem. Given the intended distance distribution, we derive in closed form the tangent density that realizes it, prove it is the only isotropic choice with chart-independent likelihoods for a broad class of charts, and prove a lower bound with explicit constants on what ignoring the problem costs a variational autoencoder. Experiments backed by an exact per-run normalization audit confirm that the compensated prior is invariant to the chart and stable across scales, every wrapped baseline we train collapses to the boundary of its chart, curvature becomes recoverable where the wrapped prior fails and protein-orientation likelihood improves from 2.58 to 0.87 nats at identical accuracy.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Information Theory (cs.IT); Differential Geometry (math.DG); Machine Learning (stat.ML)
arXiv:2511.14056 [cs.LG]
(or arXiv:2511.14056v3 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2511.14056 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 302 words
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