## 2 Materials and Methods
### The Proposed Method
\begin{array}{cccc}HFK(\mathbf{X}_{1},\mathbf{X}_{1 })&HFK(\mathbf{X}_{1},\mathbf{X}_{2})&\ldots&HFK(\mathbf{X}_{1},\mathbf{X}_{ n})\\ HFK(\mathbf{X}_{2},\mathbf{X}_{1})&HFK(\mathbf{X}_{2},\mathbf{X}_{2})&\ldots& HFK(\mathbf{X}_{2},\mathbf{X}_{n})\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ HFK(\mathbf{X}_{n},\mathbf{X}_{1})&HFK(\mathbf{X}_{n},\mathbf{X}_{2})&\ldots &HFK(\mathbf{X}_{n},\mathbf{X}_{n})\end{array} \right]. \tag{10}\]  
For any \(n\) dimensional column vector \(\mathbf{a}\), it should satisfy:  
\[\mathbf{a}^{T}\mathbf{\hat{H}}\mathbf{a}\geq 0. \tag{11}\]  
Expanding the above equation for each element, we have:  
\[\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i}a_{j}HFK(\mathbf{X}_{i},\mathbf{X}_{j})\geq 0. \tag{12}\]  
By substituting the definition of HFK (9), we can get:  
\[\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i}a_{j}exp(-\frac{t\sqrt{\|\mathbf{HF}_{i}\|_{F }^{2}\|\mathbf{HF}_{l}\|_{F}^{2}}}{vec(\mathbf{HF}_{i})^{T}vec(\mathbf{HF}_{ l})}\geq 0 \tag{13}\]  
Based on the property of the natural logarithm exponential function, the above inequality is clearly satisfied. Hence, with fulfilling the above two conditions, according to the _[PERSON]_ theorem, HFK satisfies the definition of the kernel function. After computing the similarity between all pairs of samples based on HFK function (9), the similarity matrix (also known as the adjacency matrix) \(\mathbf{S}\) can be obtained. According to the matrix \(\mathbf{S}\), the topological graph structure of the data in manifold space can also be simultaneously constructed. Following the guiding ideology of Laplacian eigenmaps, we need to find out the best low-dimensional mapping of the original data which maintains the manifold relationships in high-dimensional space. Given \(n\) samples in the original space \(\mathbf{X}=\{\mathbf{x}_{i}\cdot\cdot\cdot\mathbf{x}_{n}\}\in\mathbb{R}^{d_{ o}\times n}\). Our goal is to find out the best mapping \(\mathbf{F}=\{\mathbf{f}_{i}\cdots\mathbf{f}_{n}\}\in\mathbb{R}^{d_{t}\times n}\) in the target low-dimensional space, where \(d_{t}\ll d_{o}\). While reducing the dimensions, it is essential to preserve the relationships between data points in the original high-dimensional manifold space, therefore, an objective minimization function is constructed with the constraint of matrix \(\mathbf{S}\):  
\[Obj=\min\sum_{i,j}^{n}\parallel\mathbf{f}_{i}-\mathbf{f}_{j}\parallel_{2}^{2 }\mathbf{S}_{ij}, \tag{14}\]  
where \(||\cdot||_{2}\) is the vector norm for calculating the Euclidean distance of two vectors. The target is to minimize the value of \(Obj\). Constrained by the similarity matrix \(\mathbf{S}\), those two sample points that are close in the original high-dimensional space must also be close in the mapping of the low-dimensional space, otherwise this will result in a larger penalty. Given the definition of Laplacian matrix \(\mathbf{L}=\mathbf{D}-\mathbf{S}\), where \(\mathbf{D}\) is a diagonal matrix and its elements are defined as the sum of rows (or columns) of \(\mathbf{S}\), named the degree matrix. Then, the objective function (14) can be rewritten in the matrix form:  
\[Obj=\min Tr(\mathbf{F}^{T}\mathbf{L}\mathbf{F}),\quad s.t.\ \mathbf{F}^{T} \mathbf{D}\mathbf{F}=\mathbf{I}. \tag{15}\]  
**I** is the identity matrix. The equivalence of (14) and (15) is guaranteed by the following Proof 2. Proof. : **The equivalence of (14) and (15). ** Equation (14) can be further derived:  
\[