KAN网络可高效逼近复杂函数，理论证明其学习效率与函数平滑性正相关。
Approximation Rates in Besov Norms and Sample-Complexity of Kolmogorov-Arnold Networks with Residual Connections
- 基于Kolmogorov-Arnold定理，用可训练样条激活函数实现最优逼近
- 在任意有界或分形域上，对Besov函数的逼近率达到理论最优
- 首次给出残差KAN的样本复杂度无维度估计，适合高维光滑函数学习
受Kolmogorov-Arnold超位置定理启发，近年来出现的Kolmogorov-Arnold网络（KAN）作为深度学习主流框架的新骨干，通过可训练样条激活函数比多层感知机（MLP）更具自适应性。本文从理论上剖析了该架构：证明其可在任意有界开集或分形域 $\mathcal{X} \subset \mathbb{R}^d$ 上，以最优逼近率近似任意 Besov 函数 $f \in B^{s}_{p,q}(\mathcal{X})$，针对更弱的 Besov 范数 $B^α_{p,q}(\mathcal{X})$（其中 $α < s$）。同时，我们通过界定相关残差KAN类的伪维数，给出了统计保证。作为推论，直接得出在 $N$ 个独立同分布、无噪声样本下，学习一个具有 Besov 正则性的函数时，残差KAN模型的样本复杂度具有无维度估计，表明 KAN 可有效学习其能逼近的光滑映射。
原文摘要 · Abstract (English)
Inspired by the Kolmogorov-Arnold superposition theorem, Kolmogorov-Arnold Networks (KANs) have recently emerged as an improved backbone for most deep learning frameworks, promising more adaptivity than their multilayer perceptron (MLP) predecessor by allowing for trainable spline-based activation functions. In this paper, we probe the theoretical foundations of the KAN architecture by showing that it can optimally approximate any Besov function in $B^{s}_{p,q}(\mathcal{X})$ on a bounded open, or even fractal, domain $\mathcal{X}$ in $\mathbb{R}^d$ at the optimal approximation rate with respect to any weaker Besov norm $B^α_{p,q}(\mathcal{X})$; where $α< s$. We complement our approximation result with a statistical guarantee by bounding the pseudodimension of the relevant class of Res-KANs. As an application of the latter, we directly deduce a dimension-free estimate on the sample complexity of a residual KAN model when learning a function of Besov regularity from $N$ i.i.d. noiseless samples, showing that KANs can learn the smooth maps which they can approximate.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。