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Design of semi-tensor product-based kernel function for SVM nonlinear classification
ShengliXue1,LijunZhang2,ZeyuZhu2
0
(1 School of Mathematics and Statistics, Yulin University, Yulin 719000, Shaanxi, China;2 School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710000, Shaanxi, China)
摘要:
The kernel function method in support vector machine (SVM) is an excellent tool for nonlinear classification. How to design a kernel function is difficult for an SVM nonlinear classification problem, even for the polynomial kernel function. In this paper, we propose a new kind of polynomial kernel functions, called semi-tensor product kernel (STP-kernel), for an SVM nonlinear classification problem by semi-tensor product of matrix (STP) theory. We have shown the existence of the STP-kernel function and verified that it is just a polynomial kernel. In addition, we have shown the existence of the reproducing kernel Hilbert space (RKHS) associated with the STP-kernel function. Compared to the existing methods, it is much easier to construct the nonlinear feature mapping for an SVM nonlinear classification problem via an STP operator.
关键词:  SVM · Semi-tensor product · STP-kernel · Nonlinear classification · Reproducing kernel Hilbert space (RKHS)
DOI:https://doi.org/10.1007/s11768-022-00120-z
基金项目:This work was supported by the National Natural Science Foundation of China (61573288), the Key Programs in Shaanxi Province of China (2021JZ-12) and the Yulin Science and Technology Bureau project (2019-89-2).
Design of semi-tensor product-based kernel function for SVM nonlinear classification
Shengli Xue1,Lijun Zhang2,Zeyu Zhu2
(1 School of Mathematics and Statistics, Yulin University, Yulin 719000, Shaanxi, China;2 School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710000, Shaanxi, China)
Abstract:
The kernel function method in support vector machine (SVM) is an excellent tool for nonlinear classification. How to design a kernel function is difficult for an SVM nonlinear classification problem, even for the polynomial kernel function. In this paper, we propose a new kind of polynomial kernel functions, called semi-tensor product kernel (STP-kernel), for an SVM nonlinear classification problem by semi-tensor product of matrix (STP) theory. We have shown the existence of the STP-kernel function and verified that it is just a polynomial kernel. In addition, we have shown the existence of the reproducing kernel Hilbert space (RKHS) associated with the STP-kernel function. Compared to the existing methods, it is much easier to construct the nonlinear feature mapping for an SVM nonlinear classification problem via an STP operator.
Key words:  SVM · Semi-tensor product · STP-kernel · Nonlinear classification · Reproducing kernel Hilbert space (RKHS)