the linear transformation can then be defined as ${{\mathbf{\Lambda }}^{-1/4}}{{\mathbf{U}}^{T}}{{\mathbf{x}}_{i}}$. In literature [@hyvarinen2000independent], ${{\mathbf{\Lambda }}^{-1/2}}{{\mathbf{U}}^{T}}{{\mathbf{x}}_{i}}$ was called the PCA whitening transformation, and data whitening has been widely exploited in many applications, e.g., face recognition, object detection, and image classification [@yang2005ica; @krizhevsky2009learning; @hariharan2012discriminative; @girshick2013training]. Considering its connection with data whitening, semi-whitening is also expected to be effective in other tasks and applications.

### The subproblem on $( {\bf w}, b)$

Given ${\bf M}$, the F-SVM model can be formulated as: $$\begin{aligned}
 \label{eq:class_18}
\begin{aligned}
 \min \limits_{\mathbf{w},b,\mathbf{\xi }} \ \ \ \ & \frac{1}{2}{{\mathbf{w}}^{T}}\mathbf{Bw}+C\sum\limits_{i=1}^{n}{{{\xi }_{i}}}, \\
 s.t. \ \ \ \ & {{y}_{i}}({{\mathbf{w}}^{T}}{{\mathbf{x}}_{i}}+b)\ge 1-{{\xi }_{i}}, \forall i,\\
 \ \ \ \ & {{\xi }_{i}}\ge 0,i=1,2,\cdots ,n.
\end{aligned}\end{aligned}$$ where $\mathbf{B}={{\mathbf{M}}^{-1}}$. The eigenvalue decomposition of ${\bf B}$ is $\mathbf{B}=\mathbf{V\Sigma }{{\mathbf{V}}^{T}}$. By introducing $\mathbf{B}={{\mathbf{L}}^{T}}\mathbf{L}$, the transformation matrix ${\bf L}$ can be rewritten as to $\mathbf{L}={{\mathbf{\Sigma }}^{\frac{1}{2}}}{{\mathbf{V}}^{T}}$. Let ${{\mathbf{z}}_{i}}={{\mathbf{\Sigma }}^{-\frac{1}{2}}}{{\mathbf{V}}^{T}}{{\mathbf{x}}_{i}}$ and $\mathbf{v}=\mathbf{Lw}$. With simple algebra, the problem in Eq. (\[eq:class\_18\]) can be reformulated as: $$\begin{aligned}
 \label{eq:class_19}
\begin{aligned}
 \min \limits_{\mathbf{v},b,\mathbf{\xi }} \ \ \ \ & \frac{1}{2}{{\mathbf{v}}^{T}}\mathbf{v}+C\sum\limits_{i=1}^{n}{{{\xi }_{i}}}, \\
 s.t. \ \ \ \ & {{y}_{i}}({{\mathbf{v}}^{T}}{{\mathbf{z}}_{i}}+b)\ge 1-{{\xi }_{i}}, \forall i,\\
 \ \ \ \ & {{\xi }_{i}}\ge 0,i=1,2,\cdots ,n.
\end{aligned}\end{aligned}$$ which can be solved using the off-the-shelf SVM 