Minkowski 吸引子网络：用于视觉表示的闭式双曲流
Minkowski Attractor Networks: Closed-Form Hyperbolic Flows for Visual Representations
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中文总结 AI 辅助
本文提出 Minkowski 吸引子网络（MAN），通过闭式双曲流在伪黎曼时空嵌入视觉表示，避免黎曼优化，实现高效单次前向传播，并推出 MAN-2D 与 MAN-4D 骨干。
中文摘要 AI 辅助
几何表示学习主要将表示支撑在平坦的欧几里得子空间或紧致乘积环面（$\mathbb{T}^K$）上。然而，平坦流形具有零曲率和多项式体积增长，在嵌入多尺度、树状视觉层次时固有地遭受度量失真。虽然双曲空间（$\mathbb{H}^m$）通过恒定负曲率（$K<0$）和指数体积扩展规避了这一问题，但先前的双曲深度架构受限于计算繁琐的黎曼优化、非线性陀螺向量微积分和浮点不稳定性。在这项工作中，我们引入了 **Minkowski 吸引子网络（MAN）**，一种受算子分裂启发的框架，将表示嵌入伪黎曼 Minkowski 时空（$\mathbb{R}^{1,m}$）中。通过将双曲流形视为二次曲面水平集，MAN 通过结合线性 Lorentz 群传输与非线性锥提升和闭式径向重缩放来解析双曲几何，在单次前向传播中评估，无需数值 ODE 求解器或迭代回缩。我们建立了 **MAN-2D**（$\mathbb{R}^{1,1} \to \mathbb{H}^1$）作为我们的主要高吞吐视觉骨干，将通道因子化粒度最大化到 $D/2$ 个独立的二维 Minkowski 块。我们进一步将 **MAN-4D**（$\mathbb{R}^{1,3} \to \mathbb{H}^3$）表述为时空扩展，利用 $\mathrm{SO}^+(1,3)$ 的交换 Cartan 子代数参数化，通过两个交换的二维平面映射评估 4D Lorentz 等距，无需矩阵指数开销。
英文摘要
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.
