HYDRA: Hyperbolic Dynamic Representation Architecture for Kolmogorov-Arnold Networks

2026-08-12

AI Summary

This work addresses the high parameter redundancy, poor scalability, and low training efficiency of Kolmogorov–Arnold Networks (KANs) by introducing hyperbolic geometry into the KAN framework for the first time. The proposed method embeds inputs into the bounded hyperbolic latent space of the Poincaré ball, performs KAN-style updates in the tangent space, and incorporates a low-rank prototype module to share function transformations across hidden dimensions. By integrating spline-based function learning, a radial coordinate structure, and a radius control mechanism, the approach enhances model interpretability and training stability. Empirical evaluations on eight benchmark datasets demonstrate that the method achieves predictive performance comparable to or better than existing approaches while substantially improving parameter efficiency.

Abstract

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.

Problem

• Kolmogorov-Arnold Networks
• parameter redundancy
• scalability
• efficiency
• function approximation

Innovation

• Hyperbolic Representation
• Kolmogorov-Arnold Networks
• Parameter Efficiency
• Low-Rank Prototype
• Poincaré Ball