aiqbench analysis
This research introduces Riemannian Neural Hamiltonian Flows, a generative modeling approach that extends traditional Euclidean frameworks to curved manifolds like hyperbolic and spherical spaces. By utilizing fixed kinetic energy and an explicit geodesic leapfrog integrator, the method achieves competitive sample quality while providing new interpretability insights into how learned potentials map to target distributions. The work advances the theoretical understanding of normalizing flows by demonstrating that specific potential configurations correspond to phase-space rotations and residual memory effects across different geometric domains.
Riemannian Neural Hamiltonian Flows: Geodesic Symplectic Transport and Interpretability
arXiv:2609.21647v1 Announce Type: new Abstract: Hamiltonian normalizing flows are attractive generative models because their phase-space maps are invertible and volume preserving, but most neural constructions are formulated in Euclidean space. We introduce Riemannian Neural Hamiltonian Flows, which combine the fixed kinetic energy of a Riemannian manifold, a learned scalar potential, and an explicit geodesic leapfrog integrator. Our analysis explains how the learned Hamiltonian can be made interpretable. Every normalizable potential defines an implicit profile, and the position marginal initially accelerates along the relative score between that profile and the base. The matched potential is the interpretable specialization for which the implicit profile is the target. In the isotropic Gaussian case, the mechanism corresponds to a phase-space rotation. A local harmonic analysis extends this result around each mode of a general target on a manifold. The gap between the learned and the matched potential is the sum of a residual memory of the base and a bias of the model, and the two potentials agree when the position base has been transferred to the momentum. This can be achieved when the former is broader than the target. Numerical experiments on Euclidean, hyperbolic, and spherical spaces show competitive sample quality and numerical cost against a Riemannian continuous normalizing flow, and confirm the interpretability of the learned potential.
Abstract:Hamiltonian normalizing flows are attractive generative models because their phase-space maps are invertible and volume preserving, but most neural constructions are formulated in Euclidean space. We introduce Riemannian Neural Hamiltonian Flows, which combine the fixed kinetic energy of a Riemannian manifold, a learned scalar potential, and an explicit geodesic leapfrog integrator. Our analysis explains how the learned Hamiltonian can be made interpretable. Every normalizable potential defines an implicit profile, and the position marginal initially accelerates along the relative score between that profile and the base. The matched potential is the interpretable specialization for which the implicit profile is the target. In the isotropic Gaussian case, the mechanism corresponds to a phase-space rotation. A local harmonic analysis extends this result around each mode of a general target on a manifold. The gap between the learned and the matched potential is the sum of a residual memory of the base and a bias of the model, and the two potentials agree when the position base has been transferred to the momentum. This can be achieved when the former is broader than the target. Numerical experiments on Euclidean, hyperbolic, and spherical spaces show competitive sample quality and numerical cost against a Riemannian continuous normalizing flow, and confirm the interpretability of the learned potential.
Machine Learning (cs.LG)
arXiv:2609.21647 [cs.LG]
(or arXiv:2609.21647v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.21647 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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