$t\leftarrow \beta *t$, ${{\mathbf{M}}_{k+1}}=\mathbf{M}$, $k\leftarrow k+1$,

### Initialization of M

Because the proposed F-SVM model is convex, alternating minimization can converge to global optimum for any initialization of ${\bf M}$ and $({{\bf w}, b})$, but proper initialization is helpful in improving the computational efficiency. Thus, by further relaxing the F-SVM model in Eq. (\[eq:class\_F-SVM-2\]), we propose a semi-whitened PCA based initialization method on ${\bf M}$.

Note that ${{\mathbf{w}}^{T}}{{\mathbf{M}}^{-1}}\mathbf{w}$ is upper bounded by [@hom1991topics]: $$\begin{aligned}
 \label{eq:class_Initial_M}
\begin{aligned}
{\bf w}^{T}{{\bf M}^{-1}}{\bf w} & =tr\left( {\bf w}{\bf w}^{T}{{\bf M}^{-1}} \right)\\
 & \le \left\| {\bf w} \right\|_{2}^{2}{{\left\|{{\bf M}^{-1}} \right\|}_{2}}\\
 & \le \left\| {\bf w} \right\|_{2}^{2}{{\left\| {{{\bf M}}^{-1}} \right\|}_{*}}.
\end{aligned}\end{aligned}$$ where ${{\left\| \mathbf{A} \right\|}_{2}}$ and ${{\left\| \mathbf{A} \right\|}_{*}}$ denote the L$_2$-norm and the nuclear norm of a positive semi-definite matrix ${\bf A}$, respectively. Based on Eq. (\[eq:class\_F-SVM-2\]) and Eq. (\[eq:class\_Initial\_M\]), by setting $\mathbf{B}={{\mathbf{M}}^{-1}}$, the subproblem on ${\bf M}$ can be rewritten as the problem on ${\bf B}$ formulated as: $$\begin{aligned}
 \label{eq:class_B}
\begin{aligned}
 \min \limits_{\mathbf{B}} \ \ \ \ & L(\mathbf{B}) = {{\left\| \mathbf{B} \right\|}_{*}}+{\tau }'tr\left( {{\mathbf{B}}^{-1}}\mathbf{S} \right), \\
 s.t. \ \ \ \ & \mathbf{B}\succ 0.
\end{aligned}\end{aligned}$$ where ${\tau }'={\rho }/{{{\left\| \mathbf{w} \right\|}^{2}}}$. The eigenvalue decomposition of ${\bf S}$ is $\mathbf{S}=\mathbf{U\Lambda }{{\mathbf{U}}^{T}}$, where $\mathbf{\Lambda }=diag({{\lambda 