SW-KAN: Kolmogorov-Arnold Networks with Stieltjes-Wigert q-Orthogonal Polynomials

Amirhosein Azarpour; Seyyed Moein Kazemi

cs.LGSep 3, 2026

SW-KAN: Kolmogorov-Arnold Networks with Stieltjes-Wigert q-Orthogonal Polynomials

Authors: Amirhosein Azarpour, Seyyed Moein Kazemi

Organizations: Faculty of Mathematical Sciences, Department of Computer Science, Shahid Beheshti University, Tehran · Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran

Abstract

Kolmogorov-Arnold Networks (KANs) represent a paradigmatic shift in deep learning by replacing fixed node activations with learnable univariate functions on edges, offering enhanced interpretability and parameter efficiency. While recent polynomial-based KAN variants have addressed the computational overhead of original B-spline implementations, they introduce a fundamental yet underexplored challenge: the domain mismatch between unbounded real-valued inputs and the bounded or semi-infinite support of orthogonal polynomial bases. To address this limitation, we propose the Stieltjes-Wigert Kolmogorov-Arnold Network (SW-KAN), a novel architecture that employs Stieltjes-Wigert q-orthogonal polynomials defined on the semi-infinite domain (0, infinity). We introduce a smooth exponential-of-tanh mapping that stably bridges the domain gap while preserving well-conditioned gradients, and leverage a numerically stable three-term recurrence that evaluates polynomial expansions in O(N) operations without special-function calls. Through comprehensive experiments spanning image classification and continuous function approximation, we demonstrate that SW-KAN achieves superior accuracy-efficiency trade-offs across diverse tasks. The log-normal weight structure and learnable q-parameter of Stieltjes-Wigert polynomials provide a distinct inductive bias that enables robust performance under resource-constrained conditions, including reduced feature dimensionality and limited training data. The proposed architecture not only outperforms established polynomial KAN baselines on standard benchmarks but also exhibits strong representational capacity for approximating complex multivariate functions with remarkably few parameters, making it a compelling alternative for efficient function approximation and classification in resource-constrained settings.

Figures & tables

Panel of Figure 1 — Figure 1: Stieltjes–Wigert polynomials $S_{0}$ – $S_{5}$ for different values of $q$ .

Panel of Figure 2 — Figure 2: SW-KAN architecture for MNIST.

Model	Accuracy	#Params	Time (s)
Fermat-KAN	0.9619	105,856	757.21
AlSalam-Carlitz-KAN	0.9675	105,862	927.30
BannaiIto-KAN	0.9670	105,865	1,261.58
Boas-Buck-KAN	0.9663	105,880	1,220.49
Boubaker-KAN	0.9731	105,856	807.36
Charlier-KAN	0.9726	105,856	853.90

Table 1: Test accuracy, number of parameters, and total training time (until best epoch) for polynomial-based KAN models and SW-KAN on MNIST. (SW-KAN results are reported at the epoch of best validation accuracy, achieved within 50 epochs.)

Figure 3: Comparison of SW-KAN and standard B-spline KAN on Fashion-MNIST. Left: test accuracy and parameter counts. Right: validation accuracy during training.

Figure 4: Function approximation results for SW-KAN on three 2D surfaces. Each row corresponds to a different target function: (left) ground truth, (middle) SW-KAN prediction, (right) absolute error.

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