发表机构
Moscow Institute of Physics and Technology（莫斯科物理技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究柯尔莫哥洛夫-阿诺尔德表示定理对抗有界平移的鲁棒性，给出使用固定分段线性内层函数的近似表示的显式构造性证明，采用与具体对抗平移无关的单个外层函数。
AI 中文摘要
柯尔莫哥洛夫-阿诺尔德表示定理（KART）源自希尔伯特第13问题，近年因在神经网络（尤其是柯尔莫哥洛夫-阿诺尔德网络，KANs）中的应用重新受到关注。尽管其精确表示已明确，但该定理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KART对抗有界对抗平移的鲁棒性，给出了使用固定分段线性内层函数的近似表示的显式、自包含且构造性证明。关键在于，只要预先知道最大界，该构造采用对所有求和项保持不变、且与具体对抗平移无关的单个外层函数。
英文摘要
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.
Comments7 pages
