Kolmogorov-Arnold Networks (KANs) have emerged as a promising alternative to traditional Multi-Layer Perceptrons (MLPs), offering enhanced interpretability and a strong 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 one of optimal polyhedral region merging. We establish a rigorous polyhedral characterization of KANs, develop a complete theory of $\epsilon$-equivalent compression, and design an optimal dynamic programming algorithm that guarantees minimal compression under specified error bounds. Our theoretical analysis demonstrates that PolyKAN achieves provably minimal compression while maintaining strict error control, with polynomial-time complexity in all network parameters. The framework provides the first formal foundation for KAN compression with mathematical guarantees, opening new directions for efficient deployment of interpretable neural architectures.

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