Kolmogorov-Arnold 稳定性
Kolmogorov--Arnold stability
AI总结：
本文研究了 Kolmogorov-Arnold 表示定理在面对隐藏空间重新参数化扰动时的稳定性，并探讨了其在神经网络理论中的适用性问题。
摘要：
将 Kolmogorov 和 Arnold 的表示定理视为一种函数表示或“表达”算法，我们通过分析其对隐藏空间重新参数化扰动的稳定性来测试其鲁棒性。可以将此类重新参数化视为对手试图破坏 KA 外部函数构造的行为。我们发现 KA 在可数个连续重新参数化集合下是稳定的，但揭示了一个关于外部函数等连续性的问题，目前阻碍了取极限和对抗连续重新参数化群的进程。这一关于外部函数正则性的讨论与 KA 在神经网络一般理论中的适用性争论相关。
英文摘要：
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.
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