the proof of [**Theorem 1**]{}. In [@zhu2012learning], Zhu et al. claimed that $R=\bar{R}$ . From [**Theorem 1**]{}, $\bar R$ is only an approximation of $R$, and counter examples can be easily found to illustrate $R\ne \bar{R}$. Let $\mathbf{w}={{\mathbf{A}}^{T}}\mathbf{u}$ and $\mathbf{M}={{\mathbf{A}}^{T}}\mathbf{A}$. Since the radius $R$ is upper bounded by $\bar R$, we can approximate $R$ with $\bar R$. With simple algebra, the radius-margin SVM model in Eq. (\[eq:class\_radiu-margin-based-SVM\]) is relaxed into the following formulation: $$\begin{aligned}
 \label{eq:class_radiu-margin-based-SVM-relaxation}
\begin{aligned}
 \min_{\mathbf{w}\!,b\!,\mathbf{\xi }\!,\mathbf{M}\!,\bar{R}\!} \ \ \ \ & F(\mathbf{w},b,\mathbf{\xi },\mathbf{M},\bar{R}) \!\!=\!\! \frac{1}{2}\left( {{\mathbf{w}}^{T}}\!{{\mathbf{M}}^{-1}}\!\mathbf{w}\! \right)\bar{R}_{{}}^{2}\!+\!C\!\sum\limits_{i=1}^{n}{{{\xi }_{i}}}, \\
 s.t. \ \ \ \ &  {{y}_{i}}({{\mathbf{w}}^{T}}{{\mathbf{x}}_{i}}+b)\ge 1-{{\xi }_{i}}, \forall i,\\
 \ \ \ \ & {{\xi }_{i}}\ge 0,i=1,2,\cdots ,n, \\
 \ \ \ \ & {{({{\mathbf{x}}_{i}}-\mathbf{\bar{x}})}^{T}}\mathbf{M}({{\mathbf{x}}_{i}}-\mathbf{\bar{x}})\le {{{\bar{R}}}^{2}}.
\end{aligned}\end{aligned}$$

[**Theorem 2**]{}. The problem in Eq. (\[eq:class\_radiu-margin-based-SVM-relaxation\]) is equivalent with the following problem: $$\begin{aligned}
 \label{eq:class_radiu-margin-based-SVM-relaxation-equivalentaion}
\begin{aligned}
 \min_{\mathbf{w},b,\mathbf{\xi },\mathbf{M}} \ \ \ \ & L(\mathbf{w},b,\mathbf{\xi },\mathbf{M})\!\!=\!\!
\left\{ \frac{1}{2}\left( {{\mathbf{w}}^{T}}{{\mathbf{M}}^{-1}}\mathbf{w} \right)\!\!+\!\!C\sum\limits_{i=1}^{n}{{{\xi }_{i}}} \right\}, \\
 s.t. \ \ \ \ & {{y}_{i}}({{\mathbf{w}}^{T}}{{\mathbf{x}}_{i}}+b)\ge 1-{{\xi }_{i}}, \forall i,\\
 \ \ \ \ & {{\xi }_{i}}\ge 0, i=1,2,\cdots ,n, \\
 \ \ \ \ & {{({{\mathbf{x}}_{i}}-\mathbf{\bar{x}})}^{T}}\mathbf{M}({{\mathbf{x}}_{i}}-\mathbf{\bar{x}})\le 1, \forall i,\\
 \ \ \ \ & \mathbf{M}\succ 0.
\end{aligned}\end{aligned}$$

Denote $(\mathbf{\hat{w}},\hat{b},\mathbf{\hat{\xi }},\mathbf{\hat{M}},\hat{R})$ by the optimal solution to the 