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

Seyd Teymoor Seydi

arXiv:2406.02583v2cs.LGcs.AIcs.CV

TL;DR

The paper examines whether diverse polynomial families can serve as KAN basis functions for approximating complex functions. It surveys 18 polynomials and evaluates polynomial-basis KANs on MNIST; Gottlieb-KAN achieves the highest reported overall accuracy, Kappa, and F1 score among the listed models.

  • Problem

    The study asks how suitable different polynomial families are as KAN basis functions for approximating complex functions, evaluating them on MNIST handwritten-digit classification.

    Source: §1 · p. 2

  • Method

    The authors survey 18 polynomials by mathematical group and evaluate KAN models incorporating them on MNIST, also analyzing complexity, parameter count, training time, and performance.

    Source: §1 · p. 2

  • Results

    0.9759 overall accuracy, 0.9732 Kappa, and 0.9759 F1 score: Gottlieb-KAN achieved the highest values among the listed models.

    Source: §3 · p. 15

  • Takeaways & Limitations

    For the given MNIST task, Gottlieb-KAN may be the most suitable choice among the listed models.

    Source: §3 · p. 15

    Further investigation and tuning on more complex datasets are needed, alongside advanced analytical techniques for model selection and optimization.

    Source: §1 · p. 2

Abstract

from arXiv · show

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 .

1 Introduction

The introduction motivates KANs as interpretable alternatives for function approximation and proposes surveying 18 polynomial families as KAN basis functions. It also outlines an MNIST evaluation and analysis of model-performance factors, while calling for further work on more complex datasets.

1 Introduction

  • Prior KAN work includes arbitrary-width and -depth networks, wavelet-based Wav-KAN, smooth structurally informed KANs, and Chebyshev KAN.The introduction describes these as alternatives or developments related to KANs and neural networks.Source: §1 · p. 1
  • KANs are presented as interpretable, efficient function representations that may achieve better accuracy and generalization with fewer parameters than MLPs.The introduction also says structural knowledge may reduce training-data needs and mitigate hallucinated predictions.Source: §1 · p. 1
  • The study surveys 18 polynomials grouped by mathematical properties, including orthogonal, hypergeometric, q-, Fibonacci-related, combinatorial, and number-theoretic families.It presents a structured overview of their characteristics and potential KAN applications.Source: §1 · p. 2
  • The authors propose evaluating polynomial-basis KANs on MNIST handwritten-digit classification to assess their suitability for approximating complex functions.Source: §1 · p. 2
  • The study analyzes relationships among model complexity, parameter count, training time, and overall performance.It presents these factors as influencing KAN model performance.Source: §1 · p. 2
  • The authors identify further investigation and tuning on more complex datasets, plus advanced analytical techniques for model selection and optimization, as future work.Source: §1 · p. 2

2.1 Kolmogorov-Arnold Representation Theorem

The representation theorem provides a way to express multivariate continuous functions through compositions of univariate functions. This reduces a multivariate problem to simpler univariate transformations.

2.1 Kolmogorov-Arnold Representation Theorem

  • The Kolmogorov-Arnold representation theorem states that a continuous multivariate function can be decomposed into a sum of compositions of univariate functions.Source: §2.1 · p. 2
  • The decomposition uses inner univariate functions ϕr,s and outer univariate functions Φr.Source: §2.1 · p. 2
  • This representation reduces a multivariate problem to simpler univariate transformations.Source: §2.1 · p. 2

2.2 Kolmogorov-Arnold Network (KAN) Formulation

The formulation represents KAN computations as learned univariate transformations composed across layers, and surveys polynomial families that can serve as bases. The illustrated Al-Salam-Carlitz and Bannai-Ito polynomials show increasingly complex behavior as degree rises.

2.2 Kolmogorov-Arnold Network (KAN) Formulation

  • KANs approximate functions by parameterizing inner and outer functions with trainable coefficients.Source: §2.2 · p. 2
  • At each layer, learned univariate functions act on input components, and their post-activations are combined across neurons.The layer computation is also expressed in matrix form through a function matrix of learned functions.Source: §2.2 · pp. 2–3
  • A full KAN composes multiple layers, with each learned transformation applied to the input vector; the scalar-output case defines h(y).Source: §2.2 · p. 3
  • The survey classifies 18 polynomials by mathematical properties and application areas, including orthogonal, hypergeometric, q-, Fibonacci-related, combinatorial, and number-theoretic groups.Source: §2.2 · p. 3

2.2.1 Al-Salam-Carlitz Polynomial

  • Al-Salam-Carlitz polynomials satisfy a three-term recurrence and form an orthogonal basis defined with a weight function involving the q-Pochhammer symbol.The q-Pochhammer symbol is described as important in q-series and orthogonal polynomials.Source: §2.2.1 · pp. 3–4
  • For a = 1.0 and q = 0.5, the plotted Al-Salam-Carlitz polynomials show heightened oscillations and faster growth near the range edges as degree n increases.Source: Figure 1 · p. 4, §2.2.1 · p. 4

2.2.2 Bannai-Ito Polynomial

  • The Bannai-Ito polynomials follow a three-term recurrence with coefficient functions ρ_n and τ_n that depend on polynomial parameters.Source: §2.2.2 · p. 4
  • Bannai-Ito polynomial behavior grows more oscillatory and larger with degree; the n=4 curve fluctuates substantially while lower degrees remain more moderate.The curves intersect multiple times, with complex behavior especially near interval boundaries.Source: §2.2.2 · p. 4

2.2.3 Askey-Wilson Polynomial

The Askey-Wilson, Boas-Buck, and Boubaker subsections describe polynomial families through their mathematical constructions and compare how their plotted behavior changes with degree.

2.2.3 Askey-Wilson Polynomial

  • Askey-Wilson polynomials form a four-parameter orthogonal family defined recursively, with parameters a, b, c, d, and q and x = cos θ.Source: §2.2.3 · pp. 4–5
  • For Askey-Wilson polynomials, degree 4 grows dramatically at interval edges, while all degrees intersect near the origin and even degrees appear symmetric.Source: §2.2.3 · p. 5, Figure 3 · p. 5

2.2.4 Boas-Buck Polynomial

  • The Boas-Buck class generalizes Hermite, Laguerre, and Jacobi families, with recurrence coefficients and a generating function specified using analytic functions.Source: §2.2.4 · p. 6
  • For Boas-Buck polynomials, degree 4 grows dramatically at interval edges; even degrees appear symmetric, and all degrees intersect near the origin.Source: §2.2.4 · p. 6, Figure 4 · p. 6

2.2.5 Boubaker Polynomial

  • Boubaker polynomials are presented as orthogonal, with a three-term recurrence and initial conditions B0(x) = 1 and B1(x) = x.Source: §2.2.5 · pp. 6–7
  • Higher-degree Boubaker polynomials show more oscillations and larger amplitudes near interval boundaries; even degrees are symmetric, unlike odd degrees.Source: §2.2.5 · p. 7, Figure 5 · p. 7

2.2.6 Fermat Polynomial

These subsections characterize five polynomial families through their definitions or recurrences and describe how their plotted behavior varies with degree.

2.2.6 Fermat Polynomial

  • Fermat polynomials show increasing oscillation from degrees 0–4, with n distinct real zeros and symmetry about the y-axis.Source: §2.2.6 · p. 7, Figure 6 · p. 8

2.2.7 Gottlieb Polynomial

  • With α = 1.0, Gottlieb polynomials of degrees 0–4 show increasing oscillation and rapid growth, with n distinct real zeros and stronger fluctuations near endpoints.Source: §2.2.7 · p. 8, Figure 7 · p. 8

2.2.8 Heptanacci Polynomial

  • For n ≥ 7, the Heptanacci recurrence forms Hn from xHn−1 and the six preceding Heptanacci polynomials.Source: §2.2.8 · p. 8
  • Heptanacci polynomials show increasing oscillation and steeper slopes from degrees 1–5, with n real zeros and greater fluctuations near interval edges.Source: §2.2.8 · p. 9, Figure 8 · p. 9

2.2.9 Hexanacci Polynomial

  • For n ≥ 6, the Hexanacci recurrence forms Hn from xHn−1 and the five preceding Hexanacci polynomials.Source: §2.2.9 · p. 9
  • Hexanacci polynomials show increasing oscillation and steeper slopes from degrees 1–5, intersecting the x-axis at n distinct points and fluctuating more near boundaries.Source: §2.2.9 · p. 9, Figure 9 · p. 10

2.2.10 Meixner-Pollaczek Polynomial

  • At λ = 1.0 and ϕ = 0.79, Meixner-Pollaczek polynomials of degrees 0–4 show increasing oscillation and magnitude, with n distinct real zeros.Source: §2.2.10 · p. 9, Figure 10 · p. 10

2.2.11 Narayana Polynomial

The subsections introduce Narayana, Octanacci, Padé, and Pentanacci polynomials through their mathematical connections or construction and describe how several families vary with degree.

2.2.11 Narayana Polynomial

  • Narayana polynomials connect lattice paths and Dyck words and Catalan numbers; degrees 1–5 show rapid right-edge growth and stronger curvature.The Narayana polynomial is degree n −1.Source: §2.2.11 · p. 10, Figure 11 · p. 11

2.2.12 Octanacci Polynomial

  • Octanacci polynomials use a seven-predecessor recurrence, and degrees 1–5 show stronger endpoint oscillations and slopes as degree rises.The recurrence applies for n ≥7 and is specified with initial conditions.Source: §2.2.12 · pp. 10–11, Figure 12 · p. 11

2.2.13 Padé Polynomial

  • Padé approximants are ratios of bounded-degree polynomials constructed to match a function’s Taylor series through a given order.The numerator and denominator degrees are at most m and n, respectively.Source: §2.2.13 · p. 12

2.2.14 Pentanacci Polynomial

  • Pentanacci polynomials generalize Fibonacci polynomials and numbers; degrees 1–5 show faster growth and stronger edge oscillations as degree increases.The passage also notes applications in combinatorics and number theory.Source: §2.2.14 · p. 12

2.2.15 Tetranacci Polynomial

The Tetranacci, Tribo, and Vieta–Pell subsections describe degree-related changes in their plotted polynomials, while Charlier polynomials are characterized as discrete orthogonal polynomials.

2.2.15 Tetranacci Polynomial

  • Tetranacci polynomials show increasing growth and oscillation amplitude with degree, especially near the plotted range boundaries.Figure 14 plots degrees n = 1 to 5.Source: §2.2.15 · pp. 12–13, Figure 14 · p. 13

2.2.16 Tribo Polynomial

  • Tribo polynomials show greater variation in oscillations and slopes as degree increases, particularly near interval endpoints.Figure 15 plots degrees n = 1 to 5.Source: §2.2.16 · pp. 13–14, Figure 15 · p. 14

2.2.17 VietaPell Polynomial

  • Vieta–Pell polynomials display stronger oscillations and faster growth with increasing degree, especially near the plotted domain’s edges.Figure 16 plots degrees n = 0 to 4.Source: §2.2.17 · p. 14, Figure 16 · p. 14

2.2.18 Charlier Polynomial

  • Charlier polynomials are discrete orthogonal solutions to the Charlier differential equation; with a = 2.0, degrees 0–4 show stronger edge oscillations and growth.The passage identifies x as a non-negative integer and a > 0 as a real parameter.Source: §2.2.18 · p. 15, Figure 17 · p. 15

2.3 Respective Parameter

The study classifies 18 surveyed polynomials by mathematical properties and application areas, and lists their defining parameter sets.

2.3 Respective Parameter

  • The 18 surveyed polynomials are grouped into categories including orthogonal, hypergeometric, q-, Fibonacci-related, combinatorial, and number-theoretic polynomials, with parameter sets listed.Some polynomials have no additional parameters beyond their degree.Source: §2.3 · p. 15

3 Results

KAN models using different polynomial basis functions were evaluated on MNIST for handwritten digit classification using overall accuracy, Kappa, and F1 score. Gottlieb-KAN ranked highest on all three metrics, while the reported model comparisons also show variation in parameter counts and training times.

3 Results

  • 0.9759 overall accuracy, 0.9732 Kappa, and 0.9759 F1 score: Gottlieb-KAN achieved the highest value on each metric.Table 2 reports performance metrics for KAN models evaluated on MNIST handwritten digit classification.Source: §3 · p. 15, Table 2 · p. 16
  • Gottlieb-KAN has 219,907 parameters; Askey-KAN has 105,871 parameters and the longest training time (2,418.00 seconds), while Heptanacci-KAN is fastest (729.22 seconds).The analysis notes that parameter count and accuracy do not vary consistently across all models.Source: §3 · p. 16, Table 3 · p. 17
  • 0.9542 to 0.9759 overall accuracy across models is described as a narrow range, indicating comparable performance in this evaluation.The authors say broader understanding requires examination of model architectures, training data, optimization strategies, and other relevant metrics.Source: §3 · p. 16

4 Conclusion

On MNIST handwritten-digit classification, Gottlieb-KAN achieved the highest overall accuracy, Kappa, and F1 score. The conclusion presents the study as an initial exploration and calls for broader evaluation and analysis.

4 Conclusion

  • Gottlieb-KAN achieved the highest overall accuracy, Kappa, and F1 score in the MNIST handwritten-digit evaluation.The comparison covered KAN models incorporating the surveyed polynomials.Source: §4 · p. 17
  • The study surveyed 18 polynomials, grouped by mathematical properties, and evaluated them as KAN basis functions on MNIST handwritten-digit classification.The groups provide a structured overview of polynomial characteristics and application areas.Source: §4 · p. 17
  • The analysis examined relationships between model complexity, parameter counts, training time, and overall performance.The conclusion reports insights into these relationships without specifying their direction.Source: §4 · p. 17
  • The narrow accuracy range means further investigation and tuning on more complex datasets are needed to understand the polynomials’ capabilities in KANs.The authors frame this study as an initial exploration.Source: §4 · p. 17
  • Future work includes broader datasets, varied architectures and training settings, and analytical techniques to quantify factors’ relative importance for model selection and optimization.The proposed settings include optimization techniques and hyperparameters.Source: §4 · p. 17
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