Layer-wise Lipschitz-Product Control for Deep Kolmogorov--Arnold Network Representations of Compositionally Structured Functions

ArXi:2604.26444v1 Announce Type: new We prove that any continuous function f from [0,1]^n to R representable by a finite computation tree with N internal nodes and compositional sparsity s = O(1) admits a deep Kolmogoro-Arnold Network (KAN) representation. Each internal node is realised by a primitive KAN block with controlled block depth and Lipschitz product. The layer-wise Lipschitz product satisfies the primary domain-sensitive bound independent of the input dimension n. It simplifies to P(KAN_f) <= max(C*,1)^L_f with L_f <= c_max * N.