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| Design of semi-tensor product-based kernel function for SVM nonlinear classification |
| ShengliXue1,LijunZhang2,ZeyuZhu2 |
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| (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) |
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| 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 |
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| 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). |
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| 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) |
