\ \ &  {{y}_{i}}({{\mathbf{w}}^{T}}{{\mathbf{x}}_{i}}+b)\ge 1-{{\xi }_{i}}, \forall i,\\
 \ \ \ \ & {{\xi }_{i}}\ge 0, i=1,2,\cdots ,n, \\
 \ \ \ \ & \sum\limits_{i=1}^{n}{{{({{\mathbf{x}}_{i}}-\mathbf{\bar{x}})}^{T}}\mathbf{M}({{\mathbf{x}}_{i}}-\mathbf{\bar{x}})}\le \kappa, \\
 \ \ \ \ & \mathbf{M}\succ 0.
\end{aligned}\end{aligned}$$ By defining the scattering matrix of the training set $\mathbf{S}=\sum\nolimits_{i=1}^{n}{({{\mathbf{x}}_{i}}-\mathbf{\bar{x}}){{({{\mathbf{x}}_{i}}-\mathbf{\bar{x}})}^{T}}}$, based on the Lagrangian multiplier method [@boyd2004convex], the problem expressed in Eq. (\[eq:class\_F-SVM-1\]) can be equivalently reformulated as the following F-SVM model: $$\begin{aligned}
 \label{eq:class_F-SVM-2}
\begin{aligned}
 \min \limits_ {\mathbf{w},b,\mathbf{\xi },\mathbf{M}} \ \ \ \ & \frac{1}{2}\left( {{\mathbf{w}}^{T}}{{\mathbf{M}}^{-1}}\mathbf{w} \right)+C\sum\limits_{i=1}^{n}{{{\xi }_{i}}}+\rho tr\left( \mathbf{MS} \right), \\
s.t. \ \ \ \ & {{y}_{i}}({{\mathbf{w}}^{T}}{{\mathbf{x}}_{i}}+b)\ge 1-{{\xi }_{i}}, \forall i,\\
 \ \ \ \ & {{\xi }_{i}}\ge 0,i=1,2,\cdots ,n, \\
 \ \ \ \ & \mathbf{M}\succ 0.
\end{aligned}\end{aligned}$$ where ${\rho}$ is the regularization parameter determined by ${\kappa}$. In the following, we prove that our F-SVM model is convex.

[**Theorem 3**]{}. The F-SVM model is a convex optimization problem. The proof can be found in [**Appendix B**]{}.

![image](framework.eps){width="6.0in"}

Alternating minimization
------------------------

In this section, we propose an efficient alternating minimization algorithm to solve the proposed F-SVM model, as illustrated in Fig. \[fig:framework\]. First, a semi-whitened PCA based initialization scheme is proposed to initialize ${\bf M}$. Then, the alternating minimization algorithm is adopted by 