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

Abstract

Geometric representation learning predominantly scaffolds representations onto flat Euclidean subspaces or compact product tori ($\mathbb{T}^K$). 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 ($\mathbb{H}^m$) 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 \textbf{Minkowski Attractor Networks (MAN)}, an operator-splitting-inspired framework that embeds representations within pseudo-Riemannian Minkowski spacetime ($\mathbb{R}^{1,m}$). 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 \textbf{MAN-2D} ($\mathbb{R}^{1,1} \to \mathbb{H}^1$) as our primary, high-throughput visual backbone, which maximizes channel factorization granularity into $D/2$ independent two-dimensional Minkowski blocks. We further formulate \textbf{MAN-4D} ($\mathbb{R}^{1,3} \to \mathbb{H}^3$) as a spacetime extension, leveraging a commuting Cartan-subalgebra parameterization of $\mathrm{SO}^+(1,3)$ to evaluate 4D Lorentz isometries via two commuting 2D planar maps without matrix-exponential overhead.