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arXiv · 2504.05255

Kolmogorov--Arnold stability

Sviatoslav V. Dzhenzher·Michael H. Freedman
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Abstract

Regarding the representation theorem of Kolmogorov and Arnold (KA) as an algorithm for representing or < > 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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Sviatoslav V. Dzhenzher, Michael H. Freedman. 2025-04-07. Kolmogorov--Arnold stability. https://arxiv.org/abs/2504.05255

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