Quaternionic Extensions of Hyperbolic Toral Automorphisms
arXiv:2608.08252
2026
Architecture
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper constructs a noncommutative lift of hyperbolic toral automorphisms from integer matrices to volume-preserving diffeomorphisms of \(\mathrm{SU}(2)^2\), using compositions of free-group Nielsen automorphisms rather than naive quaternionic exponentiation. Its transferable asset is a recurrent state-transition layer with exact group structure, Haar-volume preservation, an invariant commuting submanifold, and analytically known hyperbolic rates on that submanifold. The most direct neural-network use is a quaternionic latent-state model in which a fixed or partially learned Nielsen layer supplies long-horizon structured dynamics, while an encoder and decoder handle observations.
Ideas from this paper
Unverified
2026
Represent each recurrent latent state as a pair of unit quaternions \((q_1,q_2)\in\mathrm{SU}(2)^2\), and evolve it with a composition of elementary Nielsen maps corresponding to a chosen hyperbolic matrix \(A\in\mathrm{SL}(2,\mathbb{Z})\). The layer exactly preserves the group manifold and Haar volume, preserves the commuting locus \(q_1q_2=q_2q_1\), and reproduces toral hyperbolic dynamics there, giving a structured long-horizon prior instead of an unconstrained matrix recurrence.
Useful6/10
Difficulty5/10
Novelty8/10