Kolmogorov-Arnold 分类器系统作为通用逼近器
Kolmogorov-Arnold Classifier Systems as Universal Approximators
浏览论文内容
中文总结 AI 辅助
针对高维输入下规则和参数指数增长问题，提出基于 Kolmogorov-Arnold 定理的 KACS，将规则按维度分解，规则数降至 O(mn²)，并证明其通用逼近性，参数仅用 2%-40%。
中文摘要 AI 辅助
随着输入维度 $n$ 的增长，基于规则的机器学习（如学习分类器系统（LCSs））在函数逼近方面面临根本性的可扩展性瓶颈：规则数量和参数数量都随 $n$ 呈指数增长。传统 LCSs 直接划分 $n$ 维输入空间，需要 $\mathcal{O}(m^n)$ 条规则才能充分覆盖，其中 $m$ 是每个变量的分辨率。本文通过按维度重新组织规则，打破了这一范式，其指导原则是 Kolmogorov-Arnold 表示定理：任何连续的 $n$ 维函数都可以表示为有限个一维函数的叠加。所提出的 Kolmogorov-Arnold 分类器系统（KACS）将目标函数分解为一维子问题，并为每个子问题分配专门的规则集，将最坏情况下的规则数量从 $\mathcal{O}(m^n)$ 减少到 $\mathcal{O}(mn^2)$，并用一维模型取代 $n$ 维局部模型，每条规则仅需两个参数，与 $n$ 无关。我们还提供了首个构造性证明，表明 LCS（即 KACS）是紧致域上连续函数的通用逼近器。在与直接 $n$ 维输入空间划分方法在相同条件下进行评估时，KACS 在许多设置中达到了具有竞争力的精度，同时仅使用了 2% 到 40% 的参数。我们的实现可在该 https URL 获取。
英文摘要
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.
