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

arXiv: https://arxiv.org/abs/2609.01729 · HTML: https://mlsanity.com/paper/2609.01729/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 33 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [U-KAN Makes Strong Backbone for Medical Image Segmentation and Generation](https://mlsanity.com/paper/2406.02918/) (2024)
- [Kolmogorov-Arnold Networks are Radial Basis Function Networks](https://mlsanity.com/paper/2405.06721/) (2024)
- [Wav-KAN: Wavelet Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2405.12832/) (2024)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [Kolmogorov-Arnold Networks (KANs) for Time Series Analysis](https://mlsanity.com/paper/2405.08790/) (2024)
- [Convolutional Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.13155/) (2024)
- [TKAN: Temporal Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2405.07344/) (2024)
- [A Survey on Kolmogorov-Arnold Network](https://mlsanity.com/paper/2411.06078/) (2024)
- [fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions](https://mlsanity.com/paper/2406.07456/) (2024)
- [GKAN: Graph Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.06470/) (2024)
- [On the expressiveness and spectral bias of KANs](https://mlsanity.com/paper/2410.01803/) (2024)
- [Exploring the Potential of Polynomial Basis Functions in Kolmogorov-Arnold Networks: A Comparative Study of Different Groups of Polynomials](https://mlsanity.com/paper/2406.02583/) (2024)
- [Suitability of KANs for Computer Vision: A preliminary investigation](https://mlsanity.com/paper/2406.09087/) (2024)
- [ReLU-KAN: New Kolmogorov-Arnold Networks that Only Need Matrix Addition, Dot Multiplication, and ReLU](https://mlsanity.com/paper/2406.02075/) (2024)
- [A Benchmarking Study of Kolmogorov-Arnold Networks on Tabular Data](https://mlsanity.com/paper/2406.14529/) (2024)
- [LeanKAN: A Parameter-Lean Kolmogorov-Arnold Network Layer with Improved Memory Efficiency and Convergence Behavior](https://mlsanity.com/paper/2502.17844/) (2025)
- [PRKAN: Parameter-Reduced Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.07032/) (2025)
- [AF-KAN: Activation Function-Based Kolmogorov-Arnold Networks for Efficient Representation Learning](https://mlsanity.com/paper/2503.06112/) (2025)

## Built on by (most cited first, 1 of 1)

- [SW-KAN: Kolmogorov-Arnold Networks with Stieltjes-Wigert q-Orthogonal Polynomials](https://mlsanity.com/paper/2610.00050/) (2026)

## Abstract (arXiv)

Kolmogorov--Arnold Networks (KANs) replace the fixed scalar weights of a standard network with learnable univariate functions on each edge, but existing variants still fix the basis that those functions are built from: B-splines, Chebyshev polynomials, wavelets, or Jacobi polynomials, and learn only the combination weights over it. We introduce RecKAN, which instead defines the basis itself by a second order polynomial recurrence, Rn+1(x) = (ax2+bx+c)Rn(x) + (dx+e)Rn-1(x), whose five coefficients are learned jointly with the network. We show this recurrence recovers several classical polynomial families including both kinds of Chebyshev polynomials, Fibonacci, Pell, and Jacobsthal polynomials as special cases, and prove that its degree grows linearly in n exactly on the sub-family containing all of them, giving a concrete sense in which the learned basis can move beyond any fixed classical choice. Across multiple benchmark datasets spanning image, text, biomedical time series classification, and time series forecasting, RecKAN outperforms three parameter-matched KAN baselines (Chebyshev, Jacobi, and spline based) on all classification tasks and achieves the lowest MSE on the ETTh1 forecasting benchmark. Additionally, when used as a classifier head with a convolutional backbone, RecKAN achieves higher accuracy than standard MLP heads on Fashion MNIST, CIFAR-10, and SVHN. On a synthetic function fitting benchmark it tracks a sharply oscillatory target that a parameter comparable MLP under fits. We further show that the learned recurrence coefficients are interpretable: on the task requiring the most local structure, training moves the basis away from the linear degree growth regime that contains every classical family we identify, consistent with our theoretical analysis of what that structural shift enables.