Minkowski attractor networks speed up hyperbolic visual representations
Minkowski Attractor Networks: Closed-Form Hyperbolic Flows for Visual Representations
Computer Vision and Pattern Recognition
Summary
Visual data like images often have complex, tree-like structures that are hard to represent using ordinary flat spaces. The authors introduce Minkowski Attractor Networks, a new way to embed these visual patterns into a space with constant negative curvature called hyperbolic space. Their method avoids difficult calculations used before and can be computed efficiently in one step. This makes it easier and faster to model complicated visual relationships in machine learning systems.
What this means in practice
- •For computer vision engineers: Use Minkowski Attractor Networks as a fast backbone model to embed complex visual hierarchies efficiently without iterative numerical solvers.
- •For machine learning system builders: Integrate closed-form hyperbolic flows to reduce instability and computational overhead in hyperbolic embedding layers for hierarchical data.
Authors
Zhongping Ji
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.