r$, are combined into one integrated inequality constraint: $$\begin{aligned}
 \label{eq:class_24}
\sum\limits_{i,j}\!\!{{{({{\mathbf{x}}_{i}}-{{\mathbf{x}}_{j}})}^{T}}\!D_{\sqrt{\mathbf{\mu }}}^{T}{{D}_{\sqrt{\mathbf{\mu }}}}\!({{\mathbf{x}}_{i}}-{{\mathbf{x}}_{j}})}\!=\!tr\left( D_{\sqrt{\mathbf{\mu }}}^{T}{{D}_{\sqrt{\mathbf{\mu }}}}{{\mathbf{S}}_{t}} \right)\!\le\!{\kappa }'.\end{aligned}$$ Let ${\kappa }'=4n\kappa$ and $\mathbf{M}=D_{\sqrt{\mathbf{\mu }}}^{T}{{D}_{\sqrt{\mathbf{\mu }}}}$. One can see that the integrated inequality constraint will be equivalent with that adopted in Eq. (\[eq:class\_F-SVM-1\]).

MSVM [@zhu2012learning] was developed for simultaneous learning of the linear transformation and SVM classifier, but the MSVM model is non-convex and solved using gradient projection. Moreover, although Zhu et al. [@zhu2012learning] claimed that $R=\bar{R}$, as discussed in Section 3.1, $\bar{R}$ is only a lower bound of $R$ and counter examples can be easily found to illustrate $R\ne \bar{R}$. Compared with MSVM [@zhu2012learning], the F-SVM model is convex and robust against noise and outliers, and can be efficient solved using the optimization method introduced in Section 3.2.

Kernelization of F-SVM
======================

With the incorporation of kernel principal component analysis, linear F-SVM can be extended to kernel version for nonlinear classification. First, we show that kernel SVM is equivalent to perform linear SVM in the kernel PCA space. Then, kernel F-SVM is introduced by conducting linear F-SVM in the kernel PCA space.

Let the kernel function be $K({{\mathbf{x}}_{i}},{{\mathbf{x}}_{j}})=\varphi {{({{\mathbf{x}}_{i}})}^{T}}\varphi ({{\mathbf{x}}_{j}})$, where $\varphi \left( \mathbf{x} \right)$ defines an implicit mapping from 