Layer-wise Lipschitz-Product Control for Deep Kolmogorov--Arnold Network Representations of Compositionally Structured Functions
arXiv cs.LG / 4/30/2026
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Key Points
- The paper shows that any continuous function on [0,1]^n representable by a finite computation tree with compositional sparsity s=O(1) can be expressed using a deep Kolmogorov–Arnold Network (KAN) with controlled internal block structure.
- It introduces layer-wise Lipschitz-product control via primitive KAN blocks with bounded block depth, yielding a primary domain-sensitive bound that is independent of the input dimension n.
- For common compositional operations (+, −, ×, sin, cos) with bounded inputs, the Lipschitz product bound simplifies to P(KAN) <= 1, and the paper provides associated layer-width and range bound estimates.
- The authors derive uniform approximation error bounds and show that, for sufficiently smooth functions (f in C^m), the KAN approximation achieves optimal B-spline convergence rates.
- Experiments corroborate the theoretical claims, reporting P(KAN)=1.0 on several compositionally structured benchmark functions and addressing a previously noted gap in Lipschitz control for deep KAN stacks.
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