## 3 HDELC Algorithm
### High-Dimensional Feature Space Reconstruction Process
#### 3.1.1 Construction of the Feature Space Reconstruction Matrix  
In order to solve the problem that high-dimensional data will reduce the model's computational efficiency in the model training process, it is necessary to select the feature domain before training the high-dimensional data model. The features with high relative importance are selected as feature subsets for classifier training. The optimal subset size of the feature space is selected through the construction of the feature selection space matrix. It can retain the valuable information in the original data as much as possible in the feature selection stage and improve the efficiency and accuracy of the model operation. Suppose that the input of the decision tree is the training dataset \(M\) with the number of samples \(|M|\), and \(z=[z_{1},z_{2},\ldots,z_{n}]\) represents a sample in the dataset of dimension \(n\). Meanwhile, \(X\) is the matrix representing the training dataset with size \(l\) rows and \(n\) columns, \(Y\) is the label vector of the training dataset with length \(l\), and [PERSON]Y=[y_{1},y_{2},\ldots,y_{l}]^{T}\). In the feature selection phase, the information entropy of each feature is first calculated, and the features in the original feature space are arranged in ascending order according to the information entropy. Information entropy was introduced by [PERSON] in 1948 to quantify the uncertainty or average content of information in a source. In the concept of information theory, assuming that a source given using a discrete random variable \(X=\{x_{1},x_{2},\ldots,x_{n}\}\) has a probability of occurrence \(p_{i}=P(x_{i})(i=1,2,\ldots,n)\) and \(\sum_{i=1}^{n}p_{i}=1\), the information entropy of the source \(X\) is expressed as  
[PERSON]Ent(x_{i})=\sum_{i=1}^{n}p_{i}\log_{2}\frac{1}{p_{i}}=-k\sum_{i=1}^{n}p_{i}\ln p where \(Ent(x_{i})\) is the characterization quantity for the overall characteristics of the source, it is a measure of the uncertainty of the source [PERSON] output information and the randomness of event occurrence. In classifier integration, especially when using gradient boosting methods (e.g., gradient boosting trees) for model training, information entropy based on natural logarithms can simplify gradient computation. Moreover, the derivative of the natural logarithm has a concise form, which helps to compute the gradient and perform effective optimization. For using natural logarithms to represent information entropy, where \(k=\log_{2}e\), we use the Boltzmann constant [29]. Based on the information entropy of each feature, calculate its corresponding coefficient of variability \(\alpha_{i}\) and weight \(\gamma_{i}\) as follows. \[\alpha_{i}=1-Ent(x_{i}) \tag{5}\]  
\[\gamma_{i}=\frac{\alpha_{i}}{\sum_{1}^{n}\alpha_{i}} \tag{6}\]  
The average information entropy \(Ave_{Ent}\) of each feature is calculated and used as a basis for classifying the importance of features in the feature space, as shown in Equation (7):  
\[Ave_{Ent}=\frac{\sum_{1}^{n}Ent(x_{i})}{n} \tag{7}\]  
At this point, the average information entropy is used as the feature space division threshold. This threshold divides the feature space into two major regions, with features with information entropy less significant than the average information entropy marked as \(Ave_{Ent}\) and features with information entropy higher than average marked as \(Ave_{Ent}\) and weak features, as follows:  
\[x_{i}=\left\{ \begin{aligned} & strong\ feature(x_{i}>Ave_{Ent},x_{i}\in X)\\ & weak\ feature(x_{i}<Ave_{Ent},x_{i}\in X)\end{aligned} \right. \tag{8}\]  
Strong features are often considered those with relatively low information entropy, meaning they exhibit significant regularity or certainty in the dataset. Low entropy features distinguish categories because they exhibit significantly different distribution patterns across categories or outcomes. Secondly, strong features contribute significantly to predicting or categorizing the target variable and can significantly improve the model's performance.