Input Convex Kolmogorov Arnold Networks

TL;DR

This paper introduces input convex neural networks based on Kolmogorov-Arnold representations, providing theoretical foundations and demonstrating their effectiveness in optimal transport tasks with competitive performance.

Contribution

The paper proposes a novel input convex neural network architecture using Kolmogorov-Arnold networks, including a universal approximation theorem and numerical validation.

Findings

Cubic spline-based ICKANs show convergence supported by numerical results.

The networks perform competitively with classical ICNNs in simple tests.

ICKANs effectively solve optimal transport problems with convex function approximations.

Abstract

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.