the data space to high or infinite dimensional feature space. For the training set $\mathcal{S}=\left\{ \left( {{\mathbf{x}}_{1}},{{y}_{1}} \right),...,\left( {{\mathbf{x}}_{n}},{{y}_{n}} \right) \right\}$, we use $\mathbf{W}=\left[ {{\mathbf{w}}_{1}},{{\mathbf{w}}_{2}},...,{{\mathbf{w}}_{D}} \right]$ to denote all the PCA eigenvectors corresponding to positive eigenvalues. Let $\mathbf{\bar{W}}$ be a set of basis vectors in the complementary space of $\mathbf{W}$. Assuming the training set is centered, for any ${{\bf x}_{i}}$, we have ${{\mathbf{\bar{W}}}^{T}}\varphi ({{\mathbf{x}}_{i}})=0$, and thus can get: $$\begin{aligned}
 \label{eq:class_25}
\begin{aligned}
 K({{\mathbf{x}}_{i}},{{\mathbf{x}}_{j}})&\!=\!\varphi {{({{\mathbf{x}}_{i}})}^{T}}\mathbf{W}{{\mathbf{W}}^{T}}\varphi ({{\mathbf{x}}_{j}})\!+\!\varphi {{({{\mathbf{x}}_{i}})}^{T}}\mathbf{\bar{W}}{{\mathbf{\bar{W}}}^{T}}\varphi ({{\mathbf{x}}_{j}})\\
&=\varphi {{({{\mathbf{x}}_{i}})}^{T}}\mathbf{W}{{\mathbf{W}}^{T}}\varphi ({{\mathbf{x}}_{j}}).
\end{aligned}\end{aligned}$$ Let ${{\mathbf{f}}_{i}}={{\mathbf{W}}^{T}}\varphi ({{\mathbf{x}}_{i}})$. The dual problem of SVM in the kernel PCA space can be formulated as: $$\begin{aligned}
 \label{eq:class_26}
\begin{aligned}
  \max \limits_{\mathbf{\alpha }} \ \ \ \ & Q\left( \mathbf{\alpha } \right)=\sum\nolimits_{i=1}^{n}{{{\alpha}_{i}}}-\frac{1}{2}\sum\nolimits_{i,j=1}^{n}{{{\alpha}_{i}}{{\alpha}_{j}}{{y}_{i}}{{y}_{j}}\left\langle {{\mathbf{f}}_{i}},{{\mathbf{f}}_{j}} \right\rangle }, \\
  s.t. \ \ \ \ & \sum\nolimits_{i=1}^{n}{{{\alpha }_{i}}{{y}_{i}}} = 0, \\
  \ \ \ \ & 0\le {{\alpha }_{i}}\le C,\ \ i=1,...,n.
\end{aligned}\end{aligned}$$ where $\left\langle {{\mathbf{f}}_{i}},{{\mathbf{f}}_{j}} \right\rangle =K\left( {{\mathbf{x}}_{i}},{{\mathbf{x}}_{j}} \right)$. Therefore, kernel SVM is equivalent to performing linear SVM in the kernel PCA space. To extend F-SVM to its kernelized version, we first project each training sample ${{\bf x}_i}$ to the kernel PCA space ${{\mathbf{f}}_{i}}={{\mathbf{W}}^{T}}\varphi ({{\mathbf{x}}_{i}})$, and then solve the following F-SVM model: $$\begin{aligned}
 \label{eq:class_27}
\begin{aligned}
 \min \limits_{\mathbf{w},b,\mathbf{\xi },\mathbf{M}}  \ \ \ \ &  