the tradeoff parameter. The objective function in Eq. (\[eq:class\_SVM\]) aims to maximize the margin $\gamma ={1}/{{{\left\| \mathbf{u} \right\|}^{2}}}$ while minimizing the empirical risk $\sum\nolimits_{i=1}^{n}{{{\xi }_{i}}}$. For joint learning, we introduce a linear transformation matrix ${\bf A}$ and integrate the radius information, resulting in the following radius-margin based SVM model: $$\begin{aligned}
 \label{eq:class_radiu-margin-based-SVM}
\begin{aligned}
 \min \limits_{\mathbf{u},b,{\xi},{\mathbf{A}},R} \ \ \ \ & \frac{1}{2}\left\| \mathbf{u} \right\|_{2}^{2}{R}^2+C\sum\nolimits_{i}{{{\xi }_{i}}}, \\
 s.t. \ \ \ \ & {{y}_{i}}({{\mathbf{u}}^{T}}{\bf A}{{\mathbf{x}}_{i}}+b)\ge 1-{{\xi }_{i}}, \forall i, \\
 \ \ \ \ & {{\xi }_{i}} \ge 0, i=1,2,\cdots ,n.
\end{aligned}\end{aligned}$$ where the radius $R$ is defined as: $$\begin{aligned}
 \label{eq:class_F-SVM-R}
\min_{ {R}, {\mathbf{x}}_{0} } {R} ^{2}, s.t. \| {{\bf A}{\mathbf{x}}_{i}}- {{\bf A}{\mathbf{x}}_{0}} \|_{2}^{2} \le {{R}^{2}},i=1,2,\cdots ,n.\end{aligned}$$ Note that $R^2$ depends on matrix $\bf A$ and the problem in Eq. (\[eq:class\_radiu-margin-based-SVM\]) is non-convex [@do2013convex]. Denote ${\bf {x}_{0}}$ by the center of all instances, and $\bar R$ by the largest squared distance from the center in transformed feature space. Let ${{\mathbf{x}}_{0}}=\mathbf{\bar{x}}=\sum\nolimits_{i=1}^{n}{{{\mathbf{x}}_{i}}}$ and $\bar{R}= \max_{i} \left\| {\bf A}{{\bf x}_{i}}-{\bf A}{\bf \bar{x}} \right\|_{2}^{2}$. We prove that the radius $R$ is bounded by $\bar R$.

[**Theorem 1**]{}. The radius $R$ is bounded by $\bar R$ by: $$\begin{aligned}
 \label{eq:class_R-bound}
\frac{1}{2}\overline{R}\le R\le \overline{R}.\end{aligned}$$ Please refer to [**Appendix A**]{} for 