while our F-SVM is proposed to minimize the convex relaxation of the radius-margin ratio. The generalization error is bounded by the radius and margin ratio, and the radius is determined by the spread along all possible directions rather than only the direction perpendicular to the separating hyperplane, making F-SVM theoretically more promising.

MR-SVM [@do2009feature], R-SVM$^{+}$ [@do2013convex] and R-SVM$_{\mu}^{+}$ [@do2013convex] aim to learn the diagonal feature transformation ${{\mathbf{D}}_{\mathbf{\mu }}}=Diag\left( \mathbf{\mu } \right)$ with ${{\mu }_{k}}\ge 0$, while F-SVM is developed for joint learning of feature transformation and SVM classifier. Both R-SVM$^{+}$ and R-SVM$_{\mu}^{+}$ need to solve a Quadratically Constrained Quadratic Programming (QCQP) optimization problem, which is computationally expensive than the alternating minimization method used in our F-SVM. Moreover, R-SVM$^{+}$ and R-SVM$_{\mu}^{+}$ adopted a tighter approximation ${{R}_{O}}$ of the radius. In F-SVM, a new approximation $\bar{R}$ of the radius is proposed, which is also tighter than that used in MR-SVM [@do2009feature]. Moreover, the individual inequality constraints on $\bar{R}$ are combined to improve the robustness against outliers. It is interesting to note that we have: $$\begin{aligned}
 \label{eq:class_23}
\sum\limits_{i,j}{{{({{\mathbf{x}}_{i}}-{{\mathbf{x}}_{j}})}^{T}}\!\mathbf{M}\!({{\mathbf{x}}_{i}}-{{\mathbf{x}}_{j}})}\!=\!tr\left( \mathbf{M}{{\mathbf{S}}_{t}} \right)\!=\!4n\left( tr\left( \mathbf{MS} \right) \right),\end{aligned}$$ where ${{\mathbf{S}}_{t}}=\sum\nolimits_{i,j}{({{\mathbf{x}}_{i}}-{{\mathbf{x}}_{j}}){{({{\mathbf{x}}_{i}}-{{\mathbf{x}}_{j}})}^{T}}}$. Eq. (\[eq:class\_23\]) indicates that, if all the inequality constraints on ${{R}_{O}}$, i.e., ${{\left\| {{D}_{\sqrt{\mathbf{\mu }}}}{{\mathbf{x}}_{i}}-{{D}_{\sqrt{\mathbf{\mu }}}}{{\mathbf{x}}_{j}} \right\|}^{2}}\le 