Abstract: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 Kolmogorov-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. For the standard operations {+,-,x,sin,cos} with x nodes on [0,1]-bounded inputs we obtain P(KAN) <= 1. Layer widths satisfy n_l <= n + 2 w_max * N. The uniform approximation error is bounded by N * max(C*,1)^d(f) * epsilon_Op (simplifies when C* <=1). For f in C^m we obtain optimal B-spline rates. Range bounds are also derived (B_f <= N+1 for additive trees). This addresses the gap on Lipschitz control in deep KAN stacks noted by Liu et al. (2024). Experiments confirm P(KAN)=1.0 for several compositionally structured functions.
| Comments: | 15 pages, theoretical note on layer-wise Lipschitz control for deep KANs |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2604.26444 [cs.LG] |
| (or arXiv:2604.26444v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2604.26444 arXiv-issued DOI via DataCite (pending registration) |
From: Aleksander Tankman [view email]
[v1]
Wed, 29 Apr 2026 08:55:49 UTC (17 KB)
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