用柯尔莫哥洛夫-阿诺德网络构建输入凸神经网络，可高效求解凸函数逼近与最优传输问题。
Input Convex Kolmogorov Arnold Networks
- 基于分段线性或三次样条的柯尔莫哥洛夫-阿诺德结构，实现输入凸性约束。
- 在简单测试中性能媲美经典输入凸神经网络（ICNN），三次样条版本数值验证收敛。
- 适用于需凸函数近似的最优传输任务，适合优化与生成模型研究者使用。
本文提出基于柯尔莫哥洛夫-阿诺德网络的输入凸神经网络（ICKAN）。第一个网络采用低阶分段线性函数表示，给出了通用逼近定理；第二个网络基于三次样条，仅通过数值实验支持收敛性。简单测试表明其性能可与经典输入凸神经网络（ICNN）相媲美。第二部分将该网络用于需要凸函数逼近的最优传输问题，实证其有效性。与ICNN对比，三次样条版ICKAN结果相近。
原文摘要 · Abstract (English)
This article presents an input convex neural network architecture using Kolmogorov-Arnold networks (ICKAN). Two specific networks are presented: the first is based on a low-order, linear-by-part, representation of functions, and a universal approximation theorem is provided. The second is based on cubic splines, for which only numerical results support convergence. We demonstrate on simple tests that these networks perform competitively with classical input convex neural networks (ICNNs). In a second part, we use the networks to solve some optimal transport problems needing a convex approximation of functions and demonstrate their effectiveness. Comparisons with ICNNs show that cubic ICKANs produce results similar to those of classical ICNNs.
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