用双曲空间优化KAN网络，提升效率与可解释性
HYDRA: Hyperbolic Dynamic Representation Architecture for Kolmogorov-Arnold Networks
- 将输入映射到双曲球面，用低秩原型共享函数变换
- 在8个数据集上参数更少，性能不降反升
- 双曲半径控制稳定训练，结构化坐标利于理解
Kolmogorov-Arnold网络（KAN）通过将标量权重替换为可学习的单变量函数来增强非线性函数逼近能力。然而，为每条连接分配独立函数会导致大量参数冗余，限制其可扩展性和效率。为此，我们提出超双曲动态表征架构（HYDRA），一种参数高效的KAN双曲扩展，结合样条函数学习与庞加莱球上的表示。HYDRA将向量输入映射至有界双曲隐空间，在切空间执行类似KAN的更新，并采用低秩原型块在隐藏维度间共享函数变换。所得双曲表示提供结构化的径向坐标以供解释，而半径控制则通过防止边界饱和提升训练稳定性。在八个基准数据集上的广泛实验表明，HYDRA在保持或超越竞争性能的同时，显著提升参数效率与表示可解释性。
原文摘要 · Abstract (English)
Kolmogorov-Arnold Networks (KANs) enhance nonlinear function approximation by replacing scalar weights with learnable univariate functions. However, assigning an independent function to every connection results in substantial parameter redundancy, limiting their scalability and efficiency. To reduce this redundancy, we introduce \textbf{HY}perbolic \textbf{D}ynamic \textbf{R}epresentation \textbf{A}rchitecture (HYDRA), a parameter-efficient hyperbolic extension of KAN that combines spline-based functional learning with representations in the Poincaré ball. HYDRA maps vector-valued inputs into a bounded hyperbolic latent space, performs KAN-style updates in tangent space, and employs a low-rank prototype block to share functional transformations across hidden dimensions. The resulting hyperbolic representations provide a structured radial coordinate for interpretation, while radius control improves training stability by preventing boundary saturation. Extensive experiments across eight benchmark datasets demonstrate that HYDRA consistently achieves competitive or superior predictive performance while improving parameter efficiency and representation interpretability.
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