跳转至

文章背景与核心概要

近年来，柯尔莫哥洛夫-阿诺德网络（KANs）作为传统多层感知机（MLPs）的一种有前景的替代方案受到了广泛关注。然而，标准 KAN 虽然用网络边上的可学习单变量函数代替了固定的标量权重，但它们通常依赖于固定的基函数（如 B 样条、切比雪夫多项式、小波或雅可比多项式），并且只能学习组合权重，这限制了模型在处理复杂动态时的灵活性。

为了突破这一限制，本文提出了 RecKAN。该模型的核心创新在于通过二阶多项式递归定义基函数，并将递归公式中的五个系数与网络参数进行联合学习。理论上，这种递归能够自然地恢复出多种经典的具代表性的多项式族；而在实践中，RecKAN 在图像分类、文本分类、生物医学时间序列分类以及时间序列预测等多项基准任务中，均展现出了超越同等参数量基准模型的优异性能。

RecKAN: Kolmogorov-Arnold Networks with a Learnable Recursive Polynomial Basis

RecKAN: Kolmogorov-Arnold Networks with a Learnable Recursive Polynomial Basis

Summary

Summary

RecKAN is a novel variation of Kolmogorov–Arnold Networks (KANs) that addresses a key limitation of existing models. While standard KANs replace fixed scalar weights with learnable univariate functions on network edges, they typically rely on a fixed basis (such as B-splines, Chebyshev polynomials, wavelets, or Jacobi polynomials) and only learn the combination weights.

RecKAN is a novel variation of Kolmogorov–Arnold Networks (KANs) that addresses a key limitation of existing models. While standard KANs replace fixed scalar weights with learnable univariate functions on network edges, they typically rely on a fixed basis (such as B-splines, Chebyshev polynomials, wavelets, or Jacobi polynomials) and only learn the combination weights.

RecKAN overcomes this constraint by defining the basis using a second-order polynomial recurrence: $\(R_{n+1}(x) = (ax^2+bx+c)R_n(x) + (dx+e)R_{n-1}(x)\)$ The five coefficients \((a, b, c, d, e)\) are learned jointly with the network parameters.

RecKAN overcomes this constraint by defining the basis using a second-order polynomial recurrence: $\(R_{n+1}(x) = (ax^2+bx+c)R_n(x) + (dx+e)R_{n-1}(x)\)$ The five coefficients \((a, b, c, d, e)\) are learned jointly with the network parameters.

Key Contributions & Findings

Key Contributions & Findings

  • Generalization of Classical Families: The recurrence naturally recovers several classical polynomial families as special cases, including Chebyshev polynomials (both kinds), Fibonacci, Pell, and Jacobsthal polynomials.
  • Superior Empirical Performance: RecKAN outperforms parameter-matched KAN baselines across image, text, biomedical time series classification, and time series forecasting (achieving the lowest MSE on the ETTh1 benchmark).
  • Enhanced Vision & Synthetic Tasks: When used as a classifier head with a convolutional backbone, RecKAN surpasses standard MLP heads on Fashion MNIST, CIFAR-10, and SVHN. It also successfully tracks sharply oscillatory targets in synthetic function fitting where parameter-comparable MLPs underfit.
  • Interpretability: The learned recurrence coefficients provide insight into model behavior, showing clear structural shifts away from linear degree growth when modeling complex local structures.
  • Generalization of Classical Families: The recurrence naturally recovers several classical polynomial families as special cases, including Chebyshev polynomials (both kinds), Fibonacci, Pell, and Jacobsthal polynomials.
  • Superior Empirical Performance: RecKAN outperforms parameter-matched KAN baselines across image, text, biomedical time series classification, and time series forecasting (achieving the lowest MSE on the ETTh1 benchmark).
  • Enhanced Vision & Synthetic Tasks: When used as a classifier head with a convolutional backbone, RecKAN surpasses standard MLP heads on Fashion MNIST, CIFAR-10, and SVHN. It also successfully tracks sharply oscillatory targets in synthetic function fitting where parameter-comparable MLPs underfit.
  • Interpretability: The learned recurrence coefficients provide insight into model behavior, showing clear structural shifts away from linear degree growth when modeling complex local structures.

Metadata

Metadata