基于正交多项式的核函数性质研究

摘要

The choice of kernel function and its parameters a core problem of support vector machine ( SVM) . Based on orthogonality and variability of orthogonal polynomial functions, kernel functions are constructed to be used as general kernel functions instead of some common kernels, such as polynomial kernel and Gaussian kernel. Generally, the kernel parameters are chosen only from natural number, which facilitates the kernel parameter tuning. 6 sets of orthogonal polynomial kernel functions based on Chebyshev polynomial, Legendre polynomial, Hermite polynomial, and Laguerre polynomial are discussed. The properties of these kernel functions are studied, and their robustness and generalization performance on some test datasets are compared. The obtained results provides theoretical basis and technical support for SVM classification.%核函数及其参数的选择是支持向量机( SVM)研究中的一个核心问题。正交多项式的正交性和可变性使其可以构造通用核函数以代替多项式核、高斯核等常用核函数。基于正交多项式构造核函数的参数仅在自然数中取值，因而能较大地简化核参数的选择。分析基于切比雪夫多项式、埃尔米特多项式、勒让德多项式及拉盖尔多项式构造的6类正交多项式核函数的性质，并在多个数据集上对比这些核函数的鲁棒性和泛化性，所得结论可为选择这些核函数进行支持向量分类提供理论依据和技术支持。