Support vector machine classifiers by non-Euclidean margins

In this article, the classical support vector machine (SVM) classifiers are generalized by the non-Euclidean margins. We first extend the linear models of the SVM classifiers by the non-Euclidean margins including the theorems and algorithms of the SVM classifiers by the hard margins and the soft margins. Specially, the SVM classifiers by the \begin{document}$ \infty $\end{document} -norm margins can be solved by the 1-norm optimization with sparsity. Next, we show that the non-linear models of the SVM classifiers by the \begin{document}$ q $\end{document} -norm margins can be equivalently transferred to the SVM in the \begin{document}$ p $\end{document} -norm reproducing kernel Banach spaces given by the hinge loss, where \begin{document}$ 1/p+1/q = 1 $\end{document} . Finally, we illustrate the numerical examples of artificial data and real data to compare the different algorithms of the SVM classifiers by the \begin{document}$ \infty $\end{document} -norm margin.