提出基于克莱因模型的双曲神经网络，提供新计算框架。
Klein Model for Hyperbolic Neural Networks
- 用克莱因模型构建双曲神经网络，支持直线测地线运算。
- 数值实验表明性能媲美庞加莱球模型。
- 适合需要高效几何运算的复杂结构建模任务。
双曲神经网络（HNNs）在建模复杂数据结构方面已被证明有效。然而，以往工作主要集中在庞加莱球模型和双曲面模型作为双曲空间的坐标表示，常忽视克莱因模型。尽管如此，克莱因模型因其直线测地线而具有独特优势，便于实现著名的爱因斯坦中点构造，此前已在其他模型中用于辅助HNN。本文提出基于克莱因模型的双曲神经网络框架，详细给出了该模型下有用操作的表达形式。进一步研究了克莱因线性层，证明其“切空间构造”下的标量乘法和平行传输恰好对应爱因斯坦标量乘法与爱因斯坦加法，类似于庞加莱球模型中的莫比乌斯操作。数值实验显示，克莱因HNN性能与庞加莱球模型相当，为双曲神经网络提供了第三种可选基础模块，适用于更复杂的架构构建。
原文摘要 · Abstract (English)
Hyperbolic neural networks (HNNs) have been proved effective in modeling complex data structures. However, previous works mainly focused on the Poincaré ball model and the hyperboloid model as coordinate representations of the hyperbolic space, often neglecting the Klein model. Despite this, the Klein model offers its distinct advantages thanks to its straight-line geodesics, which facilitates the well-known Einstein midpoint construction, previously leveraged to accompany HNNs in other models. In this work, we introduce a framework for hyperbolic neural networks based on the Klein model. We provide detailed formulation for representing useful operations using the Klein model. We further study the Klein linear layer and prove that the "tangent space construction" of the scalar multiplication and parallel transport are exactly the Einstein scalar multiplication and the Einstein addition, analogous to the Möbius operations used in the Poincaré ball model. We show numerically that the Klein HNN performs on par with the Poincaré ball model, providing a third option for HNN that works as a building block for more complicated architectures.
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