}_{1}},{{\lambda }_{2}},\cdots ,{{\lambda }_{d}})$ (${{\lambda }_{1}}\ge {{\lambda }_{2}}\ge \cdots \ge {{\lambda }_{d}}\ge 0$), ${{\lambda }_{i}}$ and the [*i*]{}th column of ${\bf U}$ denote the [*i*]{}th eigenvalue and eigenvector, rsespectively. With ${\bf U}$ and $\mathbf{\Lambda }$, we define $\mathbf{\hat{B}}$ as: $$\begin{aligned}
 \label{eq:class_15}
& \mathbf{\hat{B}}\!=\!\mathbf{U\Sigma}{{\mathbf{U}}^{T}}\!, \mathbf{\Sigma }\!=\!{diag}\{{{({\tau }'\!{{\lambda }_{1}})}^{\!1/\!2}},\cdots ,{{({\tau }'\!{{\lambda }_{d}})}^{\!1/\!2}}\}.\end{aligned}$$

[**Theorem 4**]{}. Given a SPD matrix ${\bf S}$ and ${\tau }'>0$, $\mathbf{\hat{B}}$ defined in Eq. (\[eq:class\_15\]) is the optimal solution to the problem: $$\begin{aligned}
 \label{eq:class_16}
\mathbf{\hat{B}}=\arg \min \limits_{\mathbf{B}} \left\{ L(\mathbf{B},{\tau }') = {{\left\| \mathbf{B} \right\|}_{*}}+{\tau }'\left( tr\left( {{\mathbf{B}}^{-1}}\mathbf{S} \right) \right) \right\}.\end{aligned}$$ The proof can be found in [**Appendix C**]{}. With $\mathbf{\hat{B}}$, the initialization of ${\bf M}$ in Eq. (\[eq:class\_F-SVM-2\]) is then defined as: $$\begin{aligned}
 \label{eq:class_17}
\begin{aligned}
 {{\bf M}_{0}} \! = \! \sqrt{{\tau }'}\mathbf{U\Xi }{{\mathbf{U}}^{T}}, \mathbf{\Xi } \!=\! {diag}\{{\left( {{\lambda }_{1}} \right)}^{\!-1\!/\!2},\cdots ,{{\left( {{\lambda }_{d}} \right)}^{\!-\!1/\!2}}\}.
\end{aligned}\end{aligned}$$ Noted that we assume that ${{\left\| \mathbf{w} \right\|}^{2}}$ is known for the initialization of ${\bf M}$. From Eq. (\[eq:class\_17\]), ${{\left\| \mathbf{w} \right\|}^{2}}$ only affects the scale factor $\sqrt{{{\tau }'}}$ to the linear transformation. Thus, we simply let ${{\left\| \mathbf{w} \right\|}^{2}}=1$ in our implementation.

It is interesting to point out that ${{\bf M}_{0}}$ in Eq. (\[eq:class\_17\]) implies a semi-whitening PCA transformation because ${{\mathbf{M}}_{0}}=\mathbf{U}{{\mathbf{\Lambda }}^{-1/2}}{{\mathbf{U}}^{T}}$, where 