Exploring the Potential of Polynomial Basis Functions in Kolmogorov-Arnold Networks: A Comparative Study of Different Groups of Polynomials

TL;DR

This study aims to investigate the suitability of these polynomials as basis functions in KAN models for complex tasks like handwritten digit classification on the MNIST dataset, and suggests the Gottlieb-KAN model achieves the highest performance across all metrics.

Abstract

This paper presents a comprehensive survey of 18 distinct polynomials and their potential applications in Kolmogorov-Arnold Network (KAN) models as an alternative to traditional spline-based methods. The polynomials are classified into various groups based on their mathematical properties, such as orthogonal polynomials, hypergeometric polynomials, q-polynomials, Fibonacci-related polynomials, combinatorial polynomials, and number-theoretic polynomials. The study aims to investigate the suitability of these polynomials as basis functions in KAN models for complex tasks like handwritten digit classification on the MNIST dataset. The performance metrics of the KAN models, including overall accuracy, Kappa, and F1 score, are evaluated and compared. The Gottlieb-KAN model achieves the highest performance across all metrics, suggesting its potential as a suitable choice for the given task. However, further analysis and tuning of these polynomials on more complex datasets are necessary to fully understand their capabilities in KAN models. The source code for the implementation of these KAN models is available at https://github.com/seydi1370/Basis_Functions .

Figures (17)

• Figure 1: Plot Al-Salam-Carlitz Polynomials with $a = 1.0$ and $q = 0.5$ for various values of $n$.
• Figure 2: Plot of Bannai-Ito Polynomials for various values of $n$ (r1 = 0.5, r2 = 0.5, s1 = 0.25, s2 = 0.25).
• Figure 3: Plot of Askey-Wilson Polynomials for various values of $n$ (a=0.5, b=0.5, c=0.5, d=0.5, q=0.5).
• Figure 4: Plot of Boas-Buck Polynomials for various values of $n$.
• Figure 5: Plot of Boubaker Polynomials for various values of $n$.