研究柯尔莫哥洛夫-阿诺德函数表示的稳定性，揭示其在重参数下的鲁棒性与局限
Kolmogorov--Arnold stability
- 将柯尔莫哥洛夫-阿诺德定理视为算法，检验其对隐藏空间重参数的稳定性
- 发现其在可数连续重参数下保持稳定，但连续群参数化时存在外函数等连续性障碍
- 结果关联神经网络理论适用性争议，适合关注函数逼近与深度学习基础的研究者
将柯尔莫哥洛夫-阿诺德（KA）表示定理视为一种函数表达算法，通过分析其对隐藏空间重参数化的鲁棒性来检验其稳定性。可将此类重参数视为试图破坏KA外函数构造的对抗行为。研究发现，KA在可数连续重参数集合下具有稳定性，但关于外函数等连续性的疑问阻碍了对连续重参数群取极限，从而无法完全抵御连续变换。该外函数正则性问题与KA在一般神经网络理论中的适用性争议密切相关。
原文摘要 · Abstract (English)
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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