Kolmogorov-Arnold Classifier Systems as Universal Approximators

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 this https URL.

Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE)

arXiv:2609.37958 [cs.LG]

(or arXiv:2609.37958v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.37958 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

https://doi.org/10.1109/TEVC.2026.3736664 Focus to learn more DOI(s) linking to related resources

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