## 3 HDELC Algorithm
### High-Dimensional Feature Space Reconstruction Process
#### 3.1.2 Feature Subspace Division and Reconstruction Process  
After constructing the feature space reconstruction matrix in the previous section, the attribute screening factor \(\beta\) is set as the screening coefficient to optimize the number of features in the set of weak feature spaces (features more significant than the threshold), and the original feature space is partitioned twice, i.e., to the set of weak feature subspaces \(X^{W}\) as follows:  
\[\left|X^{W}_{\,det}\right|=\beta\cdot\left|X^{W}\right| \tag{12}\]  
where \"\(\cdot\)\" is the product of the screening factor and each element of the matrix, \(X^{W}_{\,det}\) is the set of eliminated features in the set of weak feature space, \(\left|X^{W}_{\,det}\right|\) is the number of eliminated features in the set of weak feature space \(X^{W}\), \(\left|X^{W}\right|\) is the number of features in the set of weak feature subspace \(X^{W}\). The product of the attribute screening factor \(\beta\) and the number of features \(\left|X^{W}\right|\) in the weak feature subspace set \(X^{W}\) is the number of features that will be screened and optimized in the weak feature subspace, which corresponds to the scale of PCA downscaling.  
Let the number of features in the strong feature space be \(s\). At this point, the weak feature space is redefined as \(X^{W}\) as follows:  
\[X^{W}=\left[X^{W}*F_{D},X^{W}*(I^{\prime}-F_{D})\right] \tag{13}\]  
where \"\(*\)\" is the product of the corresponding elements of the two matrices, i.e., the Hadamard product of the two matrices, and \(I^{\prime}\) is the unit matrix of \(l\) rows \((n-s)\) columns. \(F_{D}\) denotes the weak eigenspace reconstruction matrix, which is a matrix of \(l\) rows \((n-s)\) columns, defined as follows:  
\[F_{D}=I-F_{S}=\begin{bmatrix}\mu_{1}^{\,1}&\mu_{2}^{\,1}&\cdots&\mu_{n}^{\,1 }\\ \mu_{1}^{\,2}&\mu_{2}^{\,2}&\cdots&\mu_{n}^{\,2}\\ \cdots&\cdots&\cdots&\cdots\\ \mu_{1}^{\,l}&\mu_{2}^{\,l}&\cdots&\mu_{n}^{\,l}\end{bmatrix} \tag{14}\]  
Of which,  
\[\mu_{i}^{\,j}=\begin{cases}1\big{(}i\in\{1,\ldots,n-s\},j\in\{1,\ldots,l\},x_ {i}\in X^{W}\big{)}\\ 0\big{(}i\in\{1,\ldots,n-s\},j\in\{1,\ldots,l\},x_{i}\in X^{W}_{\,det}\big{)} \end{cases} \tag{15}\]At this point, the original feature space is reconstructed as the set of strong feature spaces divided by the average information entropy and the set of weak feature spaces that are not optimized by the constraints of the attribute screening factor, defined as follows:  
\[X^{\prime}=\left[X*F_{S},X^{W}*F_{D}\right]=\left[X^{S},X^{W}-X^{W}_{del}\right] \tag{16}\]  
The attribute screening factor corresponding to the optimal accuracy in the group of trained decision tree classifiers is selected as the optimal feature space optimization coefficient. The original feature space is reconstructed as the set of strong feature spaces divided by the average information entropy and the set of the best weak feature spaces that are not constrained by the optimization coefficients to be optimized. The new feature space is constructed using the selected \(1-\left|X^{W}_{del}\right|\) principal components:  
\[X^{best}=PCA\Big{(}X,1-\left|X^{W}_{del}\right|\Big{)} \tag{17}\]  
where \(PCA\big{(}X,1-\left|X^{W}_{del}\right|\big{)}\) denotes the operation of applying PCA to the original feature space \(X\) and retaining the first \(1-\left|X^{W}_{del}\right|\) principal components.  
The reconstructed feature subspace ensemble size is used as the optimal feature subspace size and as the basis for PCA downscaling. The original feature space size is optimized using PCA downscaling as the feature ensemble size for the subsequent subspace integration work \(\left|X^{best}\right|\), as shown in Figure 2.