Degree-Optimized Cumulative Polynomial Kolmogorov-Arnold Networks

TL;DR

The paper introduces CP-KAN, a neural network architecture that uses polynomial basis functions and QUBO for efficient degree selection, showing strong performance in data-limited regression tasks and theoretical links to financial time series.

Contribution

It reformulates degree selection as a QUBO problem, enabling efficient and scalable neural network design with polynomial basis functions.

Findings

Effective in regression with limited data

Robust to input scale variations

Competitive performance across multiple domains

Abstract

We introduce cumulative polynomial Kolmogorov-Arnold networks (CP-KAN), a neural architecture combining Chebyshev polynomial basis functions and quadratic unconstrained binary optimization (QUBO). Our primary contribution involves reformulating the degree selection problem as a QUBO task, reducing the complexity from $O(D^N)$ to a single optimization step per layer. This approach enables efficient degree selection across neurons while maintaining computational tractability. The architecture performs well in regression tasks with limited data, showing good robustness to input scales and natural regularization properties from its polynomial basis. Additionally, theoretical analysis establishes connections between CP-KAN's performance and properties of financial time series. Our empirical validation across multiple domains demonstrates competitive performance compared to several traditional architectures tested, especially in scenarios where data efficiency and numerical stability are important. Our implementation, including strategies for managing computational overhead in larger networks is available in Ref.~\citep{cpkan_implementation}.