Generalized Kolmogorov–Arnold theorem for deeper KAN architectures

Develop a generalized Kolmogorov–Arnold representation theorem that corresponds to network architectures with more than two non-linear layers, thereby formalizing deeper Kolmogorov–Arnold Network compositions and their properties.

Background

The classical Kolmogorov–Arnold representation theorem underpins two-layer KANs, but the paper extends KANs to arbitrary depth without an accompanying generalized theorem.

The authors explicitly note the absence of a generalized version corresponding to deeper KANs, highlighting a foundational mathematical gap.