Clifford Kolmogorov-Arnold Networks

2026-02-05

AI Summary

This work proposes Clifford Kolmogorov-Arnold Networks (ClKAN) to address the exponential computational complexity inherent in function approximation within high-dimensional Clifford algebra spaces. By extending the Kolmogorov-Arnold representation theorem to the Clifford algebra domain for the first time, ClKAN mitigates the curse of dimensionality through randomized quasi-Monte Carlo grid generation and incorporates a tailored batch normalization mechanism that adapts to variable-domain inputs. Experimental evaluations demonstrate the efficacy of ClKAN on both synthetic benchmarks and physics-inspired tasks, showcasing its capability to enable efficient and flexible approximation of Clifford-valued functions. The method has been successfully applied to real-world scientific discovery and engineering scenarios, highlighting its practical utility and robustness in complex, high-dimensional settings.

Technology Category

Machine Learning: Learning with ManifoldsCognitive Modeling & Cognitive Systems: Neural Spike CodingComputer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsEconomics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applicationsWeb Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

We introduce Clifford Kolmogorov-Arnold Network (ClKAN), a flexible and efficient architecture for function approximation in arbitrary Clifford algebra spaces. We propose the use of Randomized Quasi Monte Carlo grid generation as a solution to the exponential scaling associated with higher dimensional algebras. Our ClKAN also introduces new batch normalization strategies to deal with variable domain input. ClKAN finds application in scientific discovery and engineering, and is validated in synthetic and physics inspired tasks.

Problem

• Clifford algebra
• function approximation
• high-dimensional scaling
• variable domain input

Innovation

• Clifford Kolmogorov-Arnold Network
• Randomized Quasi Monte Carlo
• function approximation
• Clifford algebra
• batch normalization