Kolmogorov-Arnold Classifier Systems as Universal Approximators

2026-09-29

AI Summary

This study addresses the scalability bottleneck of rule-based machine learning, where rules and parameters grow exponentially under high-dimensional inputs, by proposing the Kolmogorov-Arnold Classifier System (KACS) architecture. Leveraging the Kolmogorov-Arnold representation theorem, KACS decomposes high-dimensional functions into one-dimensional subproblems and integrates Learning Classifier Systems (LCS) to achieve efficient classification. This work provides the first constructive proof establishing LCS as a universal approximator for continuous functions, overcoming traditional limitations through dimensional recombination. Experimental results demonstrate that KACS maintains competitive accuracy while reducing parameter counts to merely 2%–40% of those required by conventional approaches, significantly enhancing model efficiency and scalability.

Abstract

As the input dimension $n$ grows, rule-based machine learning, such as Learning Classifier Systems (LCSs), faces a fundamental scalability bottleneck for function approximation: both rule count and parameter count grow exponentially with $n$. Traditional LCSs partition the $n$-dimensional input space directly, requiring $\mathcal{O}(m^n)$ rules for adequate coverage, where $m$ is the per-variable resolution. This article breaks from this paradigm by reorganizing rules dimension-wise, guided by the Kolmogorov-Arnold representation theorem: any continuous $n$-dimensional function can be expressed as a finite superposition of one-dimensional functions. The proposed Kolmogorov-Arnold Classifier System (KACS) decomposes the target function into one-dimensional subproblems and assigns a dedicated ruleset to each, reducing the worst-case rule count from $\mathcal{O}(m^n)$ to $\mathcal{O}(mn^2)$ and replacing $n$-dimensional local models with one-dimensional models requiring only two parameters per rule, independent of $n$. We also provide the first constructive proof that an LCS, namely KACS, is a universal approximator for continuous functions on compact domains. Evaluated against a direct $n$-dimensional input space partitioning approach under otherwise identical conditions, KACS achieves competitive accuracy in many settings while using only 2\% to 40\% of the parameters. Our implementation is available at https://github.com/YNU-NakataLab/KACS.

Problem

• Learning Classifier Systems
• Scalability bottleneck
• Function approximation
• Curse of dimensionality
• Universal approximator

Innovation

• Kolmogorov-Arnold Classifier System
• Learning Classifier Systems
• Universal Approximator
• Scalability
• Rule-based Machine Learning