problem in Eq. (\[eq:class\_radiu-margin-based-SVM-relaxation\]). Let $\mathbf{\tilde{M}}=\mathbf{\hat{M}}/{{\hat{R}}^{2}}$ and $\tilde{R}=1$. It is obvious to see that $(\mathbf{\hat{w}},\hat{b},\mathbf{\hat{\xi }},\mathbf{\tilde{M}},\tilde{R})$ is also the optimal solution to the problem in Eq. (\[eq:class\_radiu-margin-based-SVM-relaxation\]) because $F(\mathbf{\hat{w}},\hat{b},\mathbf{\hat{\xi }},\mathbf{\hat{M}},\hat{R})=F(\mathbf{\hat{w}},\hat{b},\mathbf{\hat{\xi }},\mathbf{\tilde{M}},\tilde{R})$.

Next we will show that $(\mathbf{\hat{w}},\hat{b},\mathbf{\hat{\xi }},\mathbf{\tilde{M}})$ is the optimal solution to the problem in Eq. (\[eq:class\_radiu-margin-based-SVM-relaxation-equivalentaion\]). If $(\mathbf{\hat{w}},\hat{b},\mathbf{\hat{\xi }},\mathbf{\tilde{M}})$ is not the optimal solution to Eq. (\[eq:class\_radiu-margin-based-SVM-relaxation-equivalentaion\]), there must exist some $({{\mathbf{w}}^{*}},{{b}^{*}},{{\mathbf{\xi }}^{*}},{{\mathbf{M}}^{*}})$ that satisfies all inequality constraints and $L({{\mathbf{w}}^{*}},{{b}^{*}},{{\mathbf{\xi }}^{*}},{{\mathbf{M}}^{*}})<L(\mathbf{\hat{w}},\hat{b},\mathbf{\hat{\xi }},\mathbf{\tilde{M}})$. Then we can define $\tilde{R}=1$ and have $F({{\mathbf{w}}^{*}},{{b}^{*}},{{\mathbf{\xi }}^{*}},{{\mathbf{M}}^{*}},\tilde{R})<F(\mathbf{\hat{w}},\hat{b},\mathbf{\hat{\xi }},\mathbf{\tilde{M}},\tilde{R})$, which is contradictory with the assumption that $(\mathbf{\hat{w}},\hat{b},\mathbf{\hat{\xi }},\mathbf{\tilde{M}},\tilde{R})$ is the optimal solution to Eq. (\[eq:class\_radiu-margin-based-SVM-relaxation\]). Thus, we can solve the problem in Eq. (\[eq:class\_radiu-margin-based-SVM-relaxation-equivalentaion\]) with the optimal solution $(\mathbf{\hat{w}},\hat{b},\mathbf{\hat{\xi }},\mathbf{\tilde{M}})$, and then obtain the optimal solution $(\mathbf{\hat{w}},\hat{b},\mathbf{\hat{\xi }},\mathbf{\tilde{M}},\tilde{R})$ to Eq. (\[eq:class\_radiu-margin-based-SVM-relaxation\]).

Without loss of generality, we assume $\tilde{R}=1$ and seek the corresponding optimal ${\bf w}$ and ${\bf M}$ by solving Eq. (\[eq:class\_radiu-margin-based-SVM-relaxation-equivalentaion\]). Moreover, to make the model robust against outliers and noisy samples, we combine the individual inequality constraints ${{({{\mathbf{x}}_{i}}-\mathbf{\bar{x}})}^{T}}\mathbf{M}({{\mathbf{x}}_{i}}-\mathbf{\bar{x}})\le 1,i=1,2,\cdots ,n$ into one integrated inequality constraint [@liu2013efficient], resulting in the following radius-margin based SVM model: $$\begin{aligned}
 \label{eq:class_F-SVM-1}
\begin{aligned}
 \min \limits_{\mathbf{w},b,\mathbf{\xi },\mathbf{M}} \ \ \ \ & \frac{1}{2}\left( {{\mathbf{w}}^{T}}{{\mathbf{M}}^{-1}}\mathbf{w} \right)+C\sum\limits_{i=1}^{n}{{{\xi }_{i}}}, \\
 s.t. \ \ 