## 4 Method
### Semi-Supervised Kernel Extreme Learning Machine Improvement Scheme
#### 4.2.2 Semi-Supervised Weighted Kernel Extreme Learning Machine Algorithm Process  
The SELMWK algorithm proposed in this study aims to use the manifold regularization method to jointly train a good-performing semi-supervised classifier on both labeled and unlabeled data. The flowchart of the algorithm is shown in Figure 2:As can be seen from Figure 2, the most significant difference between the SELMWK algorithm and the SSELM and SKELM algorithms is the mapping method of the input data to the hidden layer. SSELM uses explicit random mapping to get hidden layers. This method has high applicability but has poor operational stability and requires a large amount of computation when mapping to high dimensions. The other two methods use kernel mapping, which belongs to implicit mapping and cannot directly calculate the hidden layer. SKELM using the single-core method, has better operational stability but lower applicability. By using the weighted kernel, SELMWK can observe the data from multiple angles and extract the key information, while the applicability and operation stability are relatively good. After obtaining the hidden layer through two constraints and minimizing the manifold regularization loss function while minimizing the empirical loss function, the trained classifier is finally obtained to test the unlabeled label data. The calculation flow of the SELMWK algorithm will be described in detail below. First, define the algorithm as an optimization problem as Equation (16):  
\[\min_{\mathbf{\xi}\in\mathcal{F}}\lambda\mathrm{L}_{\mathrm{m}}(\mathbf{f}( \mathbf{X}))+\mathrm{CL}(\mathbf{Y_{L}},\mathbf{f}(\mathbf{X_{L}}))+\left\| \mathbf{f}\right\|_{\mathrm{H}}^{2} \tag{16}\]  
where \(\lambda\) and \(C\) are the penalty parameters, \(\mathrm{X}=\left[\mathrm{X}_{\mathrm{L}}^{\mathrm{T}},\mathrm{X}_{\mathrm{U}}^ {\mathrm{T}}\right]^{\mathrm{T}}\) is the input data, \(\mathrm{X}_{\mathrm{L}}\) is the labeled data, \(\mathrm{X}_{\mathrm{U}}\) is the unlabeled data, \(\mathrm{Y}_{\mathrm{L}}\) is the real label of the labeled data, \(\mathrm{f}(\cdot)\) is the correlation map. The first term is the loss function of manifold regularization, which aims to make the predicted results more in line with the geometric structure represented by the \(\mathrm{X}\) distribution of the input data. The second term is the empirical loss function, which aims to align the predicted results more with the actual results. The third term is a regularization term that prevents the model from overfitting. Because the kernel mapping is a nonlinear implicit mapping, for the convenience of expressing the point after the weighted kernel mapping of the input data \(\mathrm{X}\) is expressed as Equation (17):  
\[\mathbf{H}=\Phi(\mathbf{X}) \tag{17}\]  
In the formula, \(\mathrm{H}=\left[\mathrm{H}_{\mathrm{L}}^{\mathrm{T}},\mathrm{H}_{\mathrm{U}}^ {\mathrm{T}}\right]^{\mathrm{T}}\), \(\Phi(\cdot)\) weighted nonlinear mapping function, because \(\mathrm{K}\big{(}\mathrm{x}_{i},\mathrm{x}_{j}\big{)}=\left\langle\Phi(\mathrm{ x}_{i}),\Phi(\mathrm{x}_{j})\right\rangle\), where \(\left\langle\cdot\right\rangle\) represents the inner product of vectors, so the kernel matrix can be expressed as Equation (18):  
\[\mathbf{K}=\mathbf{H}\mathbf{H}^{\mathrm{T}} \tag{18}\]  
Figure 2: SELMWK algorithm flowchart. By using the weighted kernel, SELMWK can observe the data from multiple angles and extract the key information, while the applicability and operation stability are relatively good. After obtaining the hidden layer through two constraints and minimizing the manifold regularization loss function (Min Lm) while minimizing the empirical loss function (Min L), the trained classifier is finally obtained to test the unlabeled label data.