The Mathematics of Kolmogorov-Arnold-Networks versus Artificial Neural Networks

Abstract

This paper presents a rigorous mathematical analysis comparing Kolmogorov-Arnold-Networks (KAN) with traditional Artificial Neural Networks (ANN), establishing theoretical foundations for both approaches while highlighting their complementary strengths. We begin with a comprehensive examination of the Kolmogorov-Arnold representation theorem, providing complete proofs and establishing key connections to modern deep learning architectures. The analysis proceeds through multiple theoretical frameworks: measure theory, functional analysis, topology, and optimization theory, culminating in precise complexity bounds for both architectures. We prove several novel theoretical results, including exact convergence rates for KANs, optimal architectural trade-offs, and fundamental limits on computational efficiency. Our work reveals an essential trade-off between expressivity and computational complexity, demonstrating that while KANs achieve theoretical exactness, ANNs offer superior practical scalability. These insights suggest promising directions for hybrid architectures that could combine the theoretical guarantees of KANs with the practical advantages of ANNs.

Keywords: Kolmogorov-Arnold-Networks, Universal Approximation, Computational Complexity, Functional Analysis, Information Geometry, artificial neural networks

JEL Classification: C45, C60, C63, C56