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

Geometric representation learning predominantly scaffolds representations onto flat Euclidean subspaces or compact product tori (mathbbTK). 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 (mathbbHm) 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 Minkowski Attractor Networks (MAN), an operator-splitting-inspired framework that embeds representations within pseudo-Riemannian Minkowski spacetime (ℝ1,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 MAN-2D (ℝ1,1 → mathbbH1) as our primary, high-throughput visual backbone, which maximizes channel factorization granularity into D/2 independent two-dimensional Minkowski blocks. We further formulate MAN-4D (ℝ1,3 → mathbbH3) 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.