证明了KAN在有界扰动下仍能稳定近似表示函数。
Kolmogorov--Arnold against bounded translations
- 用固定分段线性内函数构造近似表示
- 单个外函数对所有项通用且不随扰动变化
- 为神经网络抗干扰提供理论支撑
源于希尔伯特第13问题的柯尔莫戈罗夫-阿诺德表示定理（KART）近年来因在神经网络中的应用而重获关注，特别是柯尔莫戈罗夫-阿诺德网络（KANs）。尽管精确表示已确立，其在隐藏层连续对抗扰动下的稳定性仍是关键开放问题。本文研究了KART在有界对抗平移下的鲁棒性，提供了使用固定分段线性内函数的显式、自包含且可构造的近似表示证明。关键在于，我们的构造仅需一个对所有求和项保持不变的外函数，且该外函数独立于具体对抗平移，前提是其最大边界已知。
原文摘要 · Abstract (English)
Historically originating from Hilbert's 13th problem, the Kolmogorov-Arnold representation theorem (KART) has recently experienced a major revitalisation through its applications to neural networks, specifically Kolmogorov-Arnold Networks (KANs). While the exact representation is well established, its stability under continuous adversarial perturbations of the hidden layer remains a critical open question. In this paper, we investigate the robustness of KART against bounded adversarial translations. We provide an explicit, self-contained, and constructive proof of an approximate representation using fixed, piecewise linear inner functions. Crucially, our construction employs a single outer function that remains invariant for all summands and is independent of the specific adversarial translation, provided its maximum bound is known a priori.
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