updating $( {\bf w}, b)$ and ${\bf M}$ alternatively. Fixing ${\bf M}$, the model can be reformulated as the SVM in the transformed feature space, and be solved using the off-the-shelf SVM solvers to update $( {\bf w}, b)$. Fixing $( {\bf w}, b)$, the gradient descent method [@boyd2004convex] is adopted to update the matrix ${\bf M}$. When the algorithm converges, as shown in Fig. \[fig:framework\], we can learn both a better matrix M to reduce the radius of MEB in feature space and a max-margin classifier $( {\bf w}, b)$. The alternating optimization procedure is summarized in [**Algorithm 1**]{}.

Training set $\left\{ \left( {{\mathbf{x}}_{i}},{{y}_{i}} \right)\left| \forall i \right. \right\}$. Optimal ${\bf M}$ and $( {\bf w}, b)$. $k$ = 1, Initialize\
    ${{\mathbf{M}}_{k}}=\sqrt{{{\tau }'}}\mathbf{U\Xi }{{\mathbf{U}}^{T}}$,\
    $\mathbf{\Xi }={diag}\{{{\left( {{\lambda }_{1}} \right)}^{-\frac{1}{2}}},{{\left( {{\lambda }_{2}} \right)}^{-\frac{1}{2}}},\cdots ,{{\left( {{\lambda }_{d
  }} \right)}^{-\frac{1}{2}}}\}$, // Lines 7-9: updating $({\bf w}, b)$. Do Eigenvalue decomposition on ${{\mathbf{M}}_{k}}$: ${{\mathbf{M}}_{k}}=\mathbf{V\Sigma }{{\mathbf{V}}^{T}}$, Perform linear transformation on ${{\mathbf{x}}_{i}}$: ${{\mathbf{z}}_{i}}\!\leftarrow\!{{\mathbf{\Sigma }}^{{1}/{2}}}\!{{\mathbf{V}}^{T}}\!{{\mathbf{x}}_{i}}$, Update the SVM classifier $({\bf w}, b)$ based on ${\bf Z}$, // Lines 11-18: updating ${\bf M}$. $\mathbf{M}={{\mathbf{M}}_{k}}$, ${\it t}=1$, Compute the gradient of ${\bf M}$:\
            $\nabla f(\mathbf{M})=-\frac{1}{2}\mathbf{M}_{{}}^{-1}\mathbf{w}{{\mathbf{w}}^{T}}\mathbf{M}_{{}}^{-1}+\rho \mathbf{S}$, Update ${\bf M}$: $\mathbf{M}={{\mathcal{P}}_{{{\mathcal{S}}_{+}}}}\left( \mathbf{M}-t\nabla f(\mathbf{M}) \right)$, Update the stepsize ${\it t}$: 