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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.26444 |
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| _version_ | 1866915967281397760 |
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| author | Tankman, Aleksander |
| author_facet | Tankman, Aleksander |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26444 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Layer-wise Lipschitz-Product Control for Deep Kolmogorov--Arnold Network Representations of Compositionally Structured Functions Tankman, Aleksander Machine Learning 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. |
| title | Layer-wise Lipschitz-Product Control for Deep Kolmogorov--Arnold Network Representations of Compositionally Structured Functions |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2604.26444 |