Degree-Optimized Cumulative Polynomial Kolmogorov-Arnold Networks

2025-05-21

AI Summary

This paper addresses the challenges of polynomial basis order selection—namely, difficulty in optimal choice, numerical instability, and high computational complexity—in few-shot regression. We propose CP-KAN, a novel neural architecture that employs Chebyshev polynomials as basis functions and constructs a learnable network grounded in the Kolmogorov–Arnold representation theorem. Crucially, we formulate layer-wise order selection as a single-step Quadratic Unconstrained Binary Optimization (QUBO) problem, enabling globally optimal, efficient, and low-complexity adaptive order determination. Theoretical analysis establishes a rigorous connection between CP-KAN’s performance and statistical properties of financial time series. Empirical evaluation demonstrates that CP-KAN consistently outperforms state-of-the-art baselines across diverse few-shot regression tasks, while exhibiting superior numerical stability, scale robustness, and intrinsic regularization. The implementation is publicly available.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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}.

Problem

• Optimizes degree selection in neural networks using QUBO
• Improves computational efficiency in polynomial-based architectures
• Enhances robustness for regression with limited data

Innovation

• Combines Chebyshev polynomials with QUBO optimization
• Reformulates degree selection as single-step QUBO task
• Enables efficient neuron-wise polynomial degree optimization