solvers. Given the solution ${\bf v}$, $\mathbf{w}=\mathbf{V}{{\mathbf{\Sigma }}^{-\frac{1}{2}}}\mathbf{v}$ can then be obtained.

### The subproblem on M

Given $( {\bf w}, b)$, the sub-problem on ${\bf M}$ can be reformulated as: $$\begin{aligned}
 \label{eq:class_20}
\begin{aligned}
 \min \limits_{\mathbf{M}} \ \ \ \ & f(\mathbf{M})=\frac{1}{2}\left( {{\mathbf{w}}^{T}}{{\mathbf{M}}^{-1}}\mathbf{w} \right)+\rho tr\left( \mathbf{MS} \right), \\
 s.t. \ \ \ \ & \mathbf{M}\succ 0.
\end{aligned}\end{aligned}$$ Since the objective function in Eq. (\[eq:class\_20\]) is convex and differentiable with respect to ${\bf M}$, the gradient projection method [@boyd2004convex] is adopted to update ${\bf M}$. According to [@petersen2008matrix], the gradient of $f({\bf M})$ can be obtained by: $$\begin{aligned}
\label{eq:class_21}
\nabla f(\mathbf{M})=-\frac{1}{2}{{\mathbf{M}}^{-1}}\mathbf{w}{{\mathbf{w}}^{T}}{{\mathbf{M}}^{-1}}+\rho \mathbf{S}.\end{aligned}$$ As presented in [**Algorithm 1**]{}, we use gradient projection $$\begin{aligned}
\label{eq:class_22}
\mathbf{M}={{\mathcal{P}}_{{{\mathcal{S}}_{+}}}}\left( \mathbf{M}-t\nabla f(\mathbf{M}) \right)\end{aligned}$$ to update ${\bf M}$ by choosing proper stepsize ${\it t}$ and gradually decreasing it along with iterations, where ${{\mathcal{P}}_{{{\mathcal{S}}_{+}}}}\left( \cdot  \right)$ projects a matrix onto the cone of positive semidefinite matrices.

Discussion
----------

The proposed F-SVM method has several interesting advantages while compared with the other radius-margin based SVMs, e.g., RMM [@shivaswamy2010maximum], MR-SVM [@do2009feature], R-SVM$^{+}$ [@do2013convex], R-SVM$_{\mu}^{+}$ [@do2013convex] and M-SVM [@zhu2012learning]. RMM [@shivaswamy2010maximum] is suggested to maximize the margin while restricting the spread of the data along the direction perpendicular to the separating hyperplane, 