## 3 High-Dimensional Mapping Entropy
### High-Dimensional Mapping Theory
#### 3.1.1 Kernel Function Selection  
The primary function of kernel functions is to offer a non-linear mapping that transforms low-dimensional input data into representations in high-dimensional feature spaces. This mapping partially suppresses the original data in the new high-dimensional space, while highlighting another part of the information, resulting in better separability. There is no definitive way to choose the feature space \(\mathrm{H}\) and the mapping function \(\phi\) when determining the kernel function \(K(x,z)\). Multiple feature spaces may be chosen, and within the same feature space, diverse mapping rules may be selected. The choice of the kernel function relies on domain knowledge within practical applications, and it is necessary to verify the kernel function's validity. Through the use of the mapping function \(\phi\), the kernel function \(K(x,\,z)\) is obtained through evaluating the internal product of \(\phi(x)\) and \(\phi(z)\)[25]. Revealing the underlying mapping function used in the classification process is our objective in this paper. The decision of the kernel function and the conditions it needs  
Figure 2: High-dimensional space mapping of nuclear functions. to satisfy are crucial considerations. There are several commonly used kernel functions, each with its own characteristics:  
1. Linear kernel function: Linear kernel functions are straightforward and map input vectors directly into higher-dimensional spaces. They are suitable for effectively handle linearly separable problems and data that exhibit strong linear correlations;
2. Polynomial kernel function: Polynomial kernel functions introduce non-linearity by calculating polynomial combinations of input vectors. To some extent, non-linear relations can be handled. However, the application of the polynomial kernel function needs to be determined in advance, as using too high of an order can lead to overfitting problems;
3. Gaussian (RBF) kernel function: Gaussian kernel functions are one of the most commonly used and versatile kernel functions. They are based on the concept of a Gaussian distribution and maps input vectors to infinite-dimensional feature spaces. Gaussian kernel functions have excellent non-linear modelling capabilities and can adapt well to various data distributions. The preference for Gaussian kernel functions stems from their robust non-linear modelling capabilities and adaptability. These functions generally exhibit superior performance across a multitude of machine-learning tasks. As a result, in this research paper, a mapping method for vector high-dimensional feature spaces are experimentally determined, and Gaussian radial basis kernel functions are adopted for classification using support vector machines (SVMs). The kernel mapping allows the inner product in the transformed space to be transformed into some function in the original space for computation, thus avoiding the computation of \(\phi(x)\)-specific mappings, and the inner product of the transformed high-dimensional feature space is computed using the kernel function. The kernel mapping enables the mapping of samples that are indivisible in the input space to samples that are divisible in the feature space. The steps for mapping to a high-dimensional kernel space are:  
* Take any \(N\) samples in the \(d\) dimensional sample space \(R^{d}S=\{x_{1},x_{2},\cdots,x_{N}\}\subset R^{d}\)
* Using non-linear mapping \(\Phi:R^{d}\to F,x\rightarrow\phi(x)\), map the samples in the original space \(R^{d}\) to the feature space \(F\) after the samples are \(\{\phi(x1),\phi(x2),\cdots,\phi(xN)\}\subset F\). The inner product of two functions in the eigenspace is a binary function that is reproducible when its kernel matrix is a semi-positive definite matrix [26]: \[\kappa(X_{i},X_{j})=\kappa(\cdot,X_{i})^{T}\kappa(\cdot,X_{j})\] (9) Thus, its inner product space is an endowment vector space which can be completed to obtain the Hilbert space, the regenerated kernel Hilbert space.