Input Convex Kolmogorov Arnold Networks

2025-05-27

AI Summary

To address function convex approximation—critical in optimal transport and related problems—this paper proposes Input-Convex Kolmogorov–Arnold Networks (ICKANs), the first neural architecture leveraging the Kolmogorov–Arnold representation theorem for input-convex network design. We construct two novel architectures: piecewise-linear ICKANs and differentiable cubic-spline ICKANs, both rigorously enforcing input convexity. Theoretically, we prove that piecewise-linear ICKANs possess universal approximation capability for convex functions, while cubic-spline ICKANs balance smoothness and expressive power. Empirically, ICKANs match the performance of classical Input-Convex Neural Networks (ICNNs) on convex function approximation and optimal transport benchmarks. Notably, cubic-spline ICKANs achieve accuracy on par with ICNNs in Wasserstein distance estimation, demonstrating their effectiveness, theoretical soundness, and practical utility.

Technology Category

• Search and Optimization: Non-convex Optimization
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Machine Learning: Deep Neural Architectures and Foundation Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Proposes input convex neural networks using Kolmogorov-Arnold architecture
• Solves optimal transport problems requiring convex function approximation
• Compares performance with classical input convex neural networks

Innovation

• Input convex neural network using Kolmogorov-Arnold
• Low-order linear-by-part function representation
• Cubic splines for optimal transport problems