RecKAN Introduces Learnable Recursive Polynomial Basis for KANs

This paper introduces RecKAN, an innovative extension of Kolmogorov-Arnold Networks (KANs) that significantly enhances their flexibility and performance. Traditional KANs improve upon standard neural networks by replacing fixed scalar weights with learnable univariate functions on each edge, but they still rely on pre-defined basis functions like B-splines or Chebyshev polynomials. RecKAN breaks this limitation by defining the basis itself through a second-order polynomial recurrence relation, whose five coefficients are learned dynamically alongside the network's other parameters. The authors demonstrate that this learnable recurrence can recover several classical polynomial families, including Chebyshev, Fibonacci, Pell, and Jacobsthal polynomials, as special cases. They also prove that the degree of the learned basis grows linearly in 'n' within this sub-family, providing a theoretical understanding of how RecKAN can move beyond fixed classical choices. Empirical evaluations across diverse benchmark datasets—including image classification (Fashion MNIST, CIFAR-10, SVHN), text classification, biomedical time series, and time series forecasting (ETTh1)—show that RecKAN consistently outperforms parameter-matched KAN baselines (Chebyshev, Jacobi, spline-based) on classification tasks and achieves the lowest Mean Squared Error (MSE) on forecasting. When used as a classifier head with a convolutional backbone, RecKAN also achieves higher accuracy than standard MLP heads. Furthermore, on a synthetic function fitting task, it successfully tracks sharply oscillatory targets that comparable MLPs fail to capture, highlighting its ability to learn more complex local structures.