Regarding the representation theorem of Kolmogorov and Arnold (KA) as an algorithm for representing or <<expressing>> functions, we test its robustness by analyzing its stability to withstand re-parameterizations of the hidden space. One may think of such re-parameterizations as the work of an adversary attempting to foil the construction of the KA outer function. We find KA to be stable under countable collections of continuous re-parameterizations, but unearth a question about the equi-continuity of the outer functions that, so far, obstructs taking limits and defeating continuous groups of re-parameterizations. This question on the regularity of the outer functions is relevant to the debate over the applicability of KA to the general theory of NNs.

翻译：将Kolmogorov和Arnold（KA）表示定理视为一种用于表示或“表达”函数的算法，我们通过分析其对隐空间重参数化的稳定性来检验其鲁棒性。此类重参数化可被视为试图破坏KA外函数构造的对抗性操作。我们发现KA在可数连续重参数化集合下具有稳定性，但揭示了一个关于外函数等度连续性的问题——该问题目前阻碍了极限过程的实现以及对连续重参数化群的抵御。这一关于外函数正则性的问题，对于探讨KA定理在神经网络通用理论中的适用性争议具有重要意义。