## 2 Multi-Classification LS-SVM Survey
\begin{cases}\frac{c.w_{n}}{\lambda}+\sum_{i=1}^{N}[(-x_{i}x_{i}^{T}(w_{y_{i} }-w_{n})\ -x_{i}(b_{y_{i}}-b_{n})-2x_{i})(1-a_{i})+\\ \sum_{m\
eq y_{i}}^{c}\zeta^{m}(x_{i}x_{i}^{T}(w_{y_{i}}-w_{n})+x_{i}(b_{y_{i} }-b_{m})+2x_{i})a_{i}]=0\end{cases} \]  
Eq. (8) becomes:  
\[ \begin{cases}\frac{c.b_{n}}{\lambda}+\sum_{i=1}^{N}[-(x_{i}^{T}(w_{y_{i}}-w_{n })+(b_{y_{i}}-b_{n})+2)(1-a_{i})+\\ \sum_{m\
eq y_{i}}^{c}\zeta^{m}(x_{i}^{T}(w_{y_{i}}-w_{m})\ +(b_{y_{i}}-b_{m})+2)a_{i}]=0\end{cases}  \tag{9}\]  
The above definitions allow us to manipulate Eq. (10) and rewrite it as a system of equations:  
\[\begin{cases} \begin{aligned} (C-G)W+(D-H)B&=E\\ (D-H)^{T}W+(R-Q)B&=U\end{aligned} \end{aligned} \tag{11}\]  
Solving these equations for \(W\) and \(B\), we obtain:  
\[ \begin{bmatrix}W\\ B\end{bmatrix} = \begin{bmatrix}\begin{matrix}(C-G)&(D-H)\\ (D-H)^{T}&(R-Q)\end{bmatrix} ^{-1} \begin{bmatrix}E\\ U\end{bmatrix}  \tag{12}\] \begin{table}
\begin{tabular}{l c} \hline \hline Matrix & & \\ Symbol & Matrix Element \\ \hline \(C\) & Diagonal matrix of size _(f\({}^{\bullet}\)c)_ by _(f\({}^{\bullet}\)c)_, the diagonal elements are composed of the square matrix \\  & \( c_{n}=\frac{C.I}{\lambda}+\sum_{i=1}^{N}x_{i}\,x_{i}^{T}+c\sum_{p=1}^{q(n)} \zeta^{ip}x_{ip}x_{i_{p}}^{T}\) \\ \(D\) & Diagonal matrix of size _(f\({}^{\bullet}\)c)_ by \(c\), the diagonal elements are the column vector \(d_{n}\) of length \(f\) \\  & \( d_{n}=\sum_{i=1}^{N}x_{i}\ \ +c\sum_{p=1}^{q(n)}\zeta^{ip}x_{i_{p}}\) \\ \(E\) & Column vector of size \(c\) made from \\  & \( e_{n}=2\sum_{i=1}^{N}x_{i}\ \ -2(1+c)\sum_{p=1}^{q(n)}\zeta^{i_{p}}x_{i_{p}}\) \\ \(H\) & Matrix of size _(f\({}^{\bullet}\)c)_ by \(c\). The row vector is \(h_{n}\) of length c and of the form \\  & \( h_{k}=\begin{bmatrix}\sum_{p=1}^{q(1)}\zeta^{i}x_{i_{p}}+\sum_{p=1}^{q(n)} \zeta^{i_{p}}x_{i_{p}}\\ \end{bmatrix}^{q(2)}\zeta^{2}x_{i_{p}}+\sum_{p=1}^{q(n)}\zeta^{i_{p}}x_{i_{p}}^{ T}\) &...\(\sum_{p=1}^{q(c)}\zeta^{i_{p}}x_{i_{p}}^{T}\)...\(\sum_{p=1}^{q(n)}\zeta^{i_{p}}x_{i_{p}}^{T}\) \\ \(Q\) & Square matrix of size \(c\), made from the row vector \(q_{n}\) of length \(c\) \\  & \( q_{n}=\begin{bmatrix}(q(1)+q(n))&...&(q(c)+q(n))\end{bmatrix}\) \\ \(U\) & Column vector of size \(c\), made from \(u_{n}\) such that \\  & \( u_{n}=-2(N-cq(n))\) \\ \(R\) & Square diagonal matrix of size \(c\), the diagonal elements \(r_{n}\) are as follows \\  & \( r_{n}=\frac{1}{\lambda}+N+cq(n)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Matrix Definitions. We define matrix \(A\) to be:  
\[A= \begin{bmatrix}(C-G)&(D-H)\\ (D-H)^{T}&(R-Q)\end{bmatrix} \] (13) and \[L\] to be: \[L= \begin{bmatrix}E\\ U\end{bmatrix}  \tag{14}\]  
These definitions allow us to rewrite Eq. (12) in a very compact form:  
\[ \begin{bmatrix}W\\ B\end{bmatrix} =A^{-1}L \tag{15}\]  
Eq. (15) provides the separating hyper-plane slopes and the intercept values for the different \(c\) classes. The hyper-plane parameters are uniquely defined by the matrices \(A\) and \(L\), and don't depend on the SVs or the Lagrange multipliers. A data point is tested against the decision function shown in Eq. (16) and is assigned to the class that shows the highest output value:  
\[\text{Class of x =}\arg\max_{i=1,\cdots,c}(w_{i}^{T}\,x+b_{i}) \tag{16}\]