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

arXiv: https://arxiv.org/abs/2609.37817 · HTML: https://mlsanity.com/paper/2609.37817/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 20 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 18 of 18)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [ShuffleNet V2: Practical Guidelines for Efficient CNN Architecture Design](https://mlsanity.com/paper/1807.11164/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [Fully Hyperbolic Convolutional Neural Networks for Computer Vision](https://mlsanity.com/paper/2303.15919/) (2023)
- [The Numerical Stability of Hyperbolic Representation Learning](https://mlsanity.com/paper/2211.00181/) (2022)
- [CliffordNet: All You Need is Geometric Algebra](https://mlsanity.com/paper/2601.06793/) (2026)
- [Ananke: Contractive Torus Attractor Networks](https://mlsanity.com/paper/2609.24737/) (2026)

## Built on by (most cited first, 1 of 1)

- [HySPE: Positional Encoding via Symplectic Dual Shears](https://mlsanity.com/paper/2610.10154/) (2026)

## Abstract (arXiv)

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.