提出可证明最优的KAN压缩框架，兼顾模型精简与误差控制
PolyKAN: A Polyhedral Analysis Framework for Provable and Approximately Optimal KAN Compression
- 基于KAN的分段多项式结构，将压缩建模为多面体区域合并
- 动态规划算法在指定误差下实现近似最优压缩，理论保证全局最优
- 首次为KAN压缩提供数学保障，适合可解释模型高效部署场景
Kolmogorov-Arnold网络（KANs）作为传统多层感知机（MLPs）的有前景替代方案，具备更强的可解释性与坚实的数学基础。然而，其参数效率仍是实际部署的主要挑战。本文提出PolyKAN，一种全新的理论框架，用于KAN压缩并提供模型规模缩减与近似误差的严格保证。通过利用KAN固有的分段多项式结构，将压缩问题建模为多面体区域合并任务。我们建立了KAN的严格多面体表征，发展了ε-等效压缩的完整理论，并设计了一种动态规划算法，在给定误差约束下实现近似最优压缩。理论分析表明，PolyKAN在保持严格误差控制的同时，实现了可证明的近似最优压缩，对单变量样条函数具有全局最优保证。该框架首次为KAN压缩提供了数学基础，开启了可解释神经架构高效部署的新方向。
原文摘要 · Abstract (English)
Kolmogorov-Arnold Networks (KANs) have emerged as a promising alternative to traditional Multi-Layer Perceptrons (MLPs), offering enhanced interpretability and a solid mathematical foundation. However, their parameter efficiency remains a significant challenge for practical deployment. This paper introduces PolyKAN, a novel theoretical framework for KAN compression that provides formal guarantees on both model size reduction and approximation error. By leveraging the inherent piecewise polynomial structure of KANs, we formulate the compression problem as a polyhedral region merging task. We establish a rigorous polyhedral characterization of KANs, develop a complete theory of $ε$-equivalent compression, and design a dynamic programming algorithm that achieves approximately optimal compression under specified error bounds. Our theoretical analysis demonstrates that PolyKAN achieves provably near-optimal compression while maintaining strict error control, with guaranteed global optimality for univariate spline functions. This framework provides the first formal foundation for KAN compression with mathematical guarantees, opening new directions for the efficient deployment of interpretable neural architectures.
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