## 2 Materials and Methods
### Convergence Based on Fisher Kernel Support Vector Machine  
The method proposed above is based on SVM, proving that the solution of SVM embedded with Fisher function is stable. The following two sub-problems prove 1. Firstly, the Fisher function is proved to be the kernel function of SVM. 2. The solution of the SVM using the kernel function is stable and convergent.  
1 Proof: According to [PERSON]'s theorem, the sufficient condition for the Fisher function to be a kernel function is that the Fisher matrix has to be a symmetric and semi-positive definite matrix.  
First, it needs to prove that the Fisher matrix is a symmetric matrix. The element \(K_{ij}=\kappa(x_{i},x_{j})\) of the Fisher function is defined by the inner product, then  
\[K_{ij}=\kappa(x_{i},x_{j})=x_{i}^{T}x_{j}=x_{j}^{T}x_{i}=\kappa(x_{j},x_{i})= K_{ji} \tag{17}\]  
Therefore, the Fisher function is symmetrical.  
Secondly, it needs to prove that Fisher matrix is a semi-positive definite matrix. Since the study in this paper only involves the real field, the diagonal element \(\kappa(x_{i},x_{i})=x_{i}^{T}x_{i}\geq 0\) of Fisher matrix, namely, the diagonal element K, is not negative. According to the properties of semi-positive definite matrix, the diagonal element of an n-order diagonal dominance matrix is not negative, thus making the matrix a semi-positive definite matrix. Hence, Fisher matrix \(K\big{(}X_{i},X_{j}\big{)}=\phi(X_{i})^{T}\phi(X_{j})\) is a semi-positive definite matrix.  
It can be concluded from the above proof that the Fisher function conforms to the Mercer theorem, thus making the Fisher function a kernel function.  
2 Proof: SVM with Fisher kernel function has stable local optimal solution, which indicates that the proposed method is stable.  
The dual form of the SVM using the Fisher kernel function is:  
\[\begin{split}&\min_{\alpha}\sum_{i=1}^{l}\alpha_{i}-\frac{1}{2} \sum_{i=1}^{l}\sum_{j=1}^{l}\alpha_{i}\alpha_{j}k(x_{i},x_{j})\\ & s.t.\ \alpha_{i}\geq 0,\ \sum_{i=1}^{l}\alpha_{i}y_{i}=0\end{split} \tag{18}\]  
Essentially, it is a quadratic programming issue with respect to \(\alpha=(\alpha_{1},\cdots,\alpha_{l})^{T}\), which can be obtained based on the following analysis:  
\[\min_{\alpha}\sum_{i=1}^{l}\alpha_{i}-\frac{1}{2}\sum_{i=1}^{l}\sum_{j=1}^{l} \alpha_{i}\alpha_{j}k(x_{i},x_{j})=\min_{\alpha}1^{T}\alpha-\frac{1}{2}\alpha ^{T}K\alpha \tag{19}\]  
where \(K_{ij}=\kappa(x_{i},x_{j})\). Simultaneous binding can be expressed as:  
\[\sum_{i=1}^{l}\alpha_{i}y_{i}=1\ \Leftrightarrow\ y^{T}\alpha\leq 1,-y^{T} \alpha\leq-1 \tag{20}\]  
\[\alpha_{i}\geq 0\ \Leftrightarrow\ -1^{T}\alpha\leq 0 \tag{21}\]Therefore, the dual form of the formula can be expressed as the following standard quadratic programming form:  
\[\begin{array}{l}\min_{\alpha}{}^{T}\alpha-\frac{1}{2}\alpha^{T}K\alpha\\ s.t.\ c^{T}\alpha\leq d\end{array} \tag{22}\]  
where \(c=\left(y^{T},-y^{T},-1^{T}\right)^{T}\), \(d=\left(1^{T},-1^{T},0^{T}\right)^{T}\).  
Hence, the SVM using the Fisher kernel function is a standard quadratic programming form, and same as the quadratic programming, the SVM also converges and has a stable local optimal solution.  
Therefore, the method proposed in this section is stable and convergent.