## 3 High-Dimensional Mapping Entropy
### High-Dimensional Mapping Theory
#### 3.1.2 Distance Measure of High-Dimensional Mapping Entropy  
The high-dimensional mapping entropy is computed in a high-dimensional space, whereas the sample entropy distance is computed in a two-dimensional plane, and there are some differences between the high-dimensional and two-dimensional distances when metricizing the sample distances.  
When using high-dimensional mapping entropy, a kernel-based non-linear function is employed to map the recombination vector samples to a high-dimensional kernel space by adopting the kernel method. In the high-dimensional space, the data has more dimensions and therefore the distances between data points have richer variations in more dimensions. In contrast, using only straight-line distances on a two-dimensional plane may not accurately reflect the relationships between data points. Commonly used distance metrics include Euclidean distance, Manhattan distance, Mahalanobis distance, and Chebyshev distance. The choice of a specific distance metric usually depends on the nature of the data and the needs of the application scenario:  
1. The Euclidean distance, being the most commonly used distance measurement, quantifies the straight-line separation between two points. It effectively preserves the geometric characteristics of data in a high-dimensional space, making it an ideal choice for evaluating continuous features;
2. The Manhattan distance neglects the correlation between the variables, instead concentrating solely on the aggregation of discrepancies. This can result in inaccurate representation of true differences in the data;
3. The Mahalanobis distance can produce unreliable results when the covariance matrix of the data is computationally unstable or difficult to estimate. In addition, the computational complexity of the procedure increases significantly in high-dimensional datasets;
4. The Chebyshev distance measures the difference between vectors by calculating their maximum difference in each dimension. The distance calculation potentially overlooks correlation in other dimensions, thereby neglecting critical information across multiple dimensions.  
In high-dimensional mapping entropy, we use the Euclidean distance to measure the distance between data points in a high-dimensional space. The Euclidean distance is superior in preserving the geometric structure of data in high-dimensional spaces and capturing the relationships between different dimensions. It can provide richer information including the magnitude and direction of the distance [28]. In high-dimensional spaces, discrepancies between sample points may exist across multiple dimensions. The Euclidean distance can more effectively capture these discrepancies and offer a more comprehensive representation of the data. Compared to other distance formulas, Euclidean distance calculations require only square and squaring operations, allowing for faster measurement of distances between high-dimensional sample points.  
In conclusion, the use of Euclidean distance in high-dimensional mapping entropy can better capture the multi-dimensional relationship of data and provide richer information.