Minkowski Attractor Networks: Closed-Form Hyperbolic Flows for Visual Representations

arXiv:2609.37817v1 Announce Type: new Abstract: Geometric representation learning predominantly scaffolds representations onto flat Euclidean subspaces or compact product tori (T^K). However, flat manifolds possess vanishing curvature and polynomial volume growth, inherently suffering from metric distortion when embedding multi-scale, tree-like visual hierarchies. While hyperbolic spaces (H^m) circumvent this via constant negative curvature (K<0) and exponential volume expansion, prior hyperbolic deep architectures are hindered by computationally cumbersome Riemannian optimization, non-linear gyrovector calculus, and floating-point instabilities. In this work, we introduce extbf{Minkowski Attractor Networks (MAN)}, an operator-splitting-inspired framework that embeds representations within pseudo-Riemannian Minkowski spacetime (R^{1,m}). By framing hyperbolic manifolds as quadric level sets, MAN resolves hyperbolic geometry by combining linear Lorentz group transport with non-linear cone lifting and closed-form radial rescaling, evaluating in a single forward pass without numerical ODE solvers or iterative retractions. We establish extbf{MAN-2D} (R^{1,1} o H^1) as our primary, high-throughput visual backbone, which maximizes channel factorization granularity into D/2 independent two-dimensional Minkowski blocks. We further formulate extbf{MAN-4D} (R^{1,3} o H^3) as a spacetime extension, leveraging a commuting Cartan-subalgebra parameterization of SO^+(1,3) to evaluate 4D Lorentz isometries via two commuting 2D planar maps without matrix-exponential overhead.

Source: arXiv cs.CV | 2026-09-30