Riemannian neural flows improve data modeling on curved spaces
Riemannian Neural Hamiltonian Flows: Geodesic Symplectic Transport and Interpretability
Machine Learning
Summary
This paper looks at a way to better model and generate complex data that naturally lives on curved spaces, like spheres or hyperbolic surfaces. The authors combine deep learning with ideas from geometry and physics to build flows that respect the curved structure of such spaces. Their method helps to understand and interpret the learned models, making the process more transparent. They tested their approach on different kinds of curved spaces, showing it works well and can be computed efficiently.
What this means in practice
- •For machine learning engineers: Create generative models that accurately handle data with intrinsic curved geometry, improving sampling quality and interpretability.
- •For computer graphics developers: Generate and manipulate images or shapes on curved surfaces by respecting their geometric properties through improved flow models.
Authors
Vincent Souveton
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.