用优化方法高效选择多项式阶数，提升小样本回归的稳定性和效率
Degree-Optimized Cumulative Polynomial Kolmogorov-Arnold Networks
- 将多项式阶数选择转化为可快速求解的优化问题
- 在有限数据下表现优异，对输入尺度变化不敏感
- 适合需要高数据效率和数值稳定的场景
我们提出累积多项式柯尔莫哥洛夫-阿诺德网络（CP-KAN），结合切比雪夫多项式基函数与二次无约束二值优化（QUBO）。核心贡献是将阶数选择问题重构为QUBO任务，将复杂度从$O(D^N)$降低为每层仅需一次优化。该方法可在神经元间高效选择阶数，同时保持计算可行性。架构在小样本回归任务中表现良好，对输入尺度具有鲁棒性，并具备天然正则化特性。理论分析揭示了其性能与金融时间序列特性的关联。多领域实证验证表明，相比多种传统架构，其在数据效率与数值稳定性要求高的场景中表现竞争力。实现代码及大模型计算开销管理策略见文献~\citep{cpkan_implementation}。
原文摘要 · Abstract (English)
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}.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。