Kolmogorov-Arnold Networks (KANs) offer an efficient and interpretable alternative to traditional multi-layer perceptron (MLP) architectures due to their finite network topology. However, according to the results of Kolmogorov and Vitushkin, the representation of generic smooth functions by KAN implementations using analytic functions constrained to a finite number of cutoff points cannot be exact. Hence, the convergence of KAN throughout the training process may be limited. This paper explores the relevance of smoothness in KANs, proposing that smooth, structurally informed KANs can achieve equivalence to MLPs in specific function classes. By leveraging inherent structural knowledge, KANs may reduce the data required for training and mitigate the risk of generating hallucinated predictions, thereby enhancing model reliability and performance in computational biomedicine.

翻译：科尔莫戈罗夫-阿诺德网络（KANs）凭借其有限的网络拓扑结构，为传统多层感知机（MLP）提供了高效且可解释的替代方案。然而，根据科尔莫戈罗夫和维图什金的研究结果，使用受限于有限截断点数量的解析函数实现的KANs，无法精确表示一般的平滑函数。因此，KAN在训练过程中的收敛性可能受到限制。本文探讨了平滑性在KANs中的相关性，提出结构化的平滑KANs可以在特定函数类中达到与MLP等价的效果。通过利用固有的结构知识，KANs有望减少训练所需数据量，并降低产生幻象预测的风险，从而在计算生物医学领域提升模型的可靠性与性能。