证明深度KAN网络可高效表示组合结构函数，且每层光滑性可控。
Layer-wise Lipschitz-Product Control for Deep Kolmogorov--Arnold Network Representations of Compositionally Structured Functions
- 通过分层控制李普希茨乘积，实现对深层KAN的光滑性约束。
- 在[0,1]输入下，李普希茨乘积恒为1，逼近误差随节点数线性增长。
- 适用于含加减乘三角函数的组合结构函数，特别适合高维建模。
我们证明，任何由包含N个内部节点、组合稀疏性s=O(1)的有限计算树表示的连续函数f: [0,1]^n → R，均可被深度柯尔莫哥洛夫-阿诺德网络（KAN）表示。每个内部节点由具有受控块深度和李普希茨乘积的原始KAN模块实现。分层李普希茨乘积满足与输入维度无关的主域敏感界，简化后得P(KAN_f) ≤ max(C*,1)^L_f，其中L_f ≤ c_max·N。对于标准运算{+,-,×,sin,cos}且输入x∈[0,1]时，P(KAN) ≤ 1。层宽满足n_l ≤ n + 2w_max·N。统一逼近误差上界为N·max(C*,1)^d(f)·ε_Op（当C*≤1时简化）。若f∈C^m，则达到最优B样条逼近率。还推导出值域界：对加法树有B_f ≤ N+1。该工作填补了刘等（2024）指出的深层KAN堆栈中李普希茨控制空白。实验验证了多个组合结构函数的P(KAN)=1.0。
原文摘要 · Abstract (English)
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.
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