demonstrated to be very effective in improving the classification performance. In model selection, radius-margin bound has been applied for choosing tradeoff parameter and scaling factors of SVM and $L_{1}$-SVM [@chapelle2002choosing]. In multiple kernel learning (MKL) [@gai2010learning; @do2009margin; @liu2013efficient] and feature reweighting [@weston2000feature], several variants of radius had been developed.

This paper aims to jointly learn SVM together with feature transformation by minimizing the radius-margin ratio, i.e., radius-margin based SVM, and more detailed review is given on this topic. Except [@zhu2012learning], most existing approaches [@do2009feature; @do2013convex; @schultz2004learning] require the transformation matrix to be diagonal, i.e., feature reweighting and selection. Direct use of radius-margin ratio ${R}^{2} \| {\mathbf{w}} \|_{2}^{2}$ in SVM results in a non-convex optimization problem, which makes the learning algorithm computationally expensive and unstable. By restricting the feature transformation to be diagonal ${{\mathbf{D}}_{\bf{\mu }}}\!=\!Diag\left( \bf{\mu } \right)$ with ${{\mu }_{k}} \!\ge\! 0$, Do et al. [@do2009feature] suggested that the radius is bounded with ${{\max }_{k}}\ {{\mu }_{k}}R_{k}^{2}\le R_{\mathbf{\mu }}^{2}\le \sum\nolimits_{k}{{{\mu }_{k}}R_{k}^{2}}$, where ${{R}_{k}}$ is the radius on dimension ${k}$. By approximating $R_{\mathbf{\mu }}^{2}$ with its upper bound $\sum\nolimits_{k}{{{\mu }_{k}}R_{k}^{2}}$, MR-SVM in [@do2009feature] solved the following convex relaxation problem: $$\begin{aligned}
 \label{eq:class_MR-SVM}
\begin{aligned}
 \min \limits_{ {\bf w},{b},{\mathbf \xi},{\mathbf \mu} } \ \ 