AI 中文总结
该研究提出基于q-厄米特I多项式的新型q正交核，其在20个基准数据集上的性能与经典核及其他正交多项式核相当，且数值稳定性和计算简便性更优，为核设计提供了新方向。
AI 中文摘要
支持向量机（SVM）的性能关键取决于核函数的选择，核函数可将数据隐式映射到高维特征空间。尽管径向基函数（RBF）等经典核仍被广泛使用，但正交多项式核提供了具有数学可解释性的替代方案，可融入结构化先验知识。本研究扩展了正交多项式核范式，引入基于离散q-厄米特I多项式的新型核族，这类q正交多项式通过形变参数q对经典厄米特多项式进行了推广。我们正式定义了q-厄米特核，并证明其在默瑟定理下的有效性；该核固有的有界性可自然防止湮灭和爆炸效应，无需显式缩放机制。在20个基准数据集上开展的大量实验表明，所提出的核与经典核及其他正交多项式核相比具有竞争力，同时在数值稳定性和计算简便性方面展现优势。我们的结果证实，q正交多项式是核设计的一个有前景方向，它将数学优美性与实际机器学习应用相结合，提供了可进一步扩展到新兴量子计算范式的概念和算法资源。为便于完全复现，我们在开放获取的GitHub仓库中提供了完整实现和实验流程，链接为此处的https URL。
英文摘要
The performance of Support Vector Machines (SVMs) critically depends on the kernel function choice, which enables implicit mapping of data into high-dimensional feature spaces. While classical kernels like Radial Basis Function (RBF) remain popular, orthogonal polynomial kernels offer mathematically interpretable alternatives that can incorporate structured prior knowledge. This work extends the orthogonal polynomial kernel paradigm by introducing a novel family based on discrete $q$-Hermite I polynomials, a class of $q$-orthogonal polynomials that generalize classical Hermite polynomials through a deformation parameter $q$. We formally define the q-Hermite kernel and establish its validity under Mercer's theorem. The kernel's inherent boundedness properties naturally prevent annihilation and explosion effects without requiring explicit scaling mechanisms. Extensive experiments across 20 benchmark datasets demonstrate that the proposed kernel achieves competitive performance compared to both classical kernels and other orthogonal polynomial kernels, while offering advantages in numerical stability and computational simplicity. Our results confirm that $q$-orthogonal polynomials constitute a promising direction for kernel design, bridging mathematical elegance with practical machine learning applications, that provides conceptual and algorithmic resources that may be further extended to emerging quantum computing paradigms. To facilitate full reproducibility, we provide the complete implementation and experimental pipeline in an open-access GitHub repository at https://github.com/Kokechacho/SVMs-QSVMs.
Comments27 pages, 4 figures, 9 tables. Source code and experimental pipeline available at https://github.com/Kokechacho/SVMs-QSVMs
