from three aspects: $(i)$ Instead of feature reweighting and selection, F-SVM can learn feature transformation and classifier simultaneously; $(ii)$ F-SVM adopts a new approximation for the radius of MEB in feature space; $(iii)$ In F-SVM, individual inequality constraints are combined into one holistic inequality constraint to improve the robustness against outliers. All these make F-SVM very promising for joint learning of feature transformation and SVM classifier, and the results also validate the effectiveness of F-SVM.

Radius-margin based Support Vector Machine
==========================================

Problem Formulation
-------------------

Denote $\mathcal{S}=\left\{ \left( {{\mathbf{x}}_{1}},{{y}_{1}} \right),...,\left( {{\mathbf{x}}_{n}},{{y}_{n}} \right) \right\}$ by a training set, where ${{\mathbf{x}}_{i}} \in {{\mathbf{R}}^{d}}$ and ${{y}_{i}}\in \left\{ -1,+1 \right\}$ denote the [*i*]{}th training sample and the corresponding class label, respectively. By introducing the slack variables ${{\xi }_{i}}\left( i=1,2,...,n \right)$, SVM aims to find the optimal separating hyperplane by solving the following optimization problem: $$\begin{aligned}
 \label{eq:class_SVM}
\begin{aligned}
 \min \limits_{\mathbf{u},b,{\xi}} \ \ \ \ & \frac{1}{2}\left\| \mathbf{u} \right\|_{2}^{2}+C\sum\nolimits_{i}{{{\xi }_{i}}}, \\
 s.t. \ \ \ \ & {{y}_{i}}({{\mathbf{u}}^{T}}{{\mathbf{x}}_{i}}+b)\ge 1-{{\xi }_{i}}, \forall i, \\
 \ \ \ \ & {{\xi }_{i}} \ge 0, i=1,2,\cdots ,n.
\end{aligned}\end{aligned}$$ where $({\bf u},b)$ are the parameters to describe the learned hyperplane ${{\mathbf{u}}^{T}}\mathbf{x}+b=0$, ${\xi}_{i}$ denotes the [*i*]{}th slack variable, and $C$ stands for 