# Deep-Learning-Based Identification of Broad-Absorption Line Quasars
## 3 Method
### Data Dimensionality Reduction
#### 3.1.4 Data Transformation and CNN Input  
In the method presented in this paper, the feature dimensions resulting from the different dimensionality reduction techniques (PCA, t-SNE, LLE, and ISOMAP) were inconsistent. To make the dimensionality-reduced results compatible with the CNN input  
Figure 2: Two-dimensional representation of the BAL quasar (blue) and non-BAL quasar (red) data after reduction using the t-SNE algorithm.  
format (a fixed length of 1426), we designed an extension method to map the reduced dimensions to uniform lengths while retaining as much information as possible from the original reduced results.  
The result of PCA dimensionality reduction was a 100-dimensional feature set, representing the principal components that captured the most information from the original spectrum. However, CNN requires input data to have a fixed length of 1426. To adapt the PCA result to the CNN input format, the 100-dimensional features were extended through linear interpolation, treating each principal component as an independent one-dimensional feature sequence. Linear interpolation was applied to each sequence, expanding the data points from 100 to 1426. This method maintained the smoothness of the feature distribution during interpolation while avoiding excessive information loss. After interpolation, all extended principal components were combined along the time steps to form a single-channel input sequence length of 1426. This process ensured the integrity of the dimensionality-reduced features while adapting to the CNN architecture's requirement for fixed-length inputs.  
The result of a t-SNE dimensionality reduction is typically 2D, intended to represent the nested relationships in high-dimensional data. To make this compatible with the CNN input format, we applied a repeating padding method to extend the 2D features generated by t-SNE, filling each dimension until the data points in each dimension reached 1426. The repeating padding method effectively extended the feature length while avoiding the potential information distortion caused by the interpolation. After extension, the two-dimensional features were combined into a single-channel input sequence length of 1426. The padding and combination process ensured the independence and consistency of the feature information, providing a sufficient representation of the nested relationships for the classification task.  
The results of LLE and ISOMAP dimensionality reductions are typically 20-dimensional, capturing the local linear relationships or global manifold structures of the original data. Using a combination of interpolation and padding, each feature sequence from each dimension was first extended through linear interpolation, expanding the data points from 20 to 1426. If the number of interpolated points was less than 1426, the length was completed by circular padding or repetition. After interpolation and padding, all dimensional features were combined along the time steps to form a single-channel input sequence length of 1426. This method retained the key feature information from the dimensionality-reduced results while ensuring consistency with the CNN input format.