Hyperbolic neural networks (HNNs) have been proved effective in modeling complex data structures. However, previous works mainly focused on the Poincar\'e 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 M\"obius operations used in the Poincar\'e ball model. We show numerically that the Klein HNN performs on par with the Poincar\'e ball model, providing a third option for HNN that works as a building block for more complicated architectures.

翻译：双曲神经网络（HNNs）已被证明在建模复杂数据结构方面具有显著效果。然而，先前的研究主要集中在将庞加莱球模型和双曲面模型作为双曲空间的坐标表示，往往忽视了Klein模型。尽管如此，Klein模型凭借其直线测地线特性而具有独特优势，这有助于实现著名的爱因斯坦中点构造——该构造先前已在其他模型中与双曲神经网络结合使用。本文提出了一种基于Klein模型的双曲神经网络框架。我们详细阐述了使用Klein模型表示关键运算的数学表述。进一步研究了Klein线性层，并证明了标量乘法和平行移动的"切空间构造"本质上就是爱因斯坦标量乘法与爱因斯坦加法，这与庞加莱球模型中使用的默比乌斯运算具有类比关系。数值实验表明，Klein双曲神经网络的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的构建模块，可用于构建更复杂的网络架构。