## 2 Materials and Methods
### The Proposed Method
The heat kernel, proposed in Laplacian eigenmaps [23], is widely adopted for estimating manifold similarities, which are defined as:  
\[s(\mathbf{x}_{i},\mathbf{x}_{l})=\mathbf{exp}(-\frac{||\mathbf{x}_{i}-\mathbf{ x}_{l}||_{F}^{2}}{t}), \tag{6}\]  
where \(\mathbf{exp}(x)\) refers to \(e^{(x)}\) and \(e\) is the natural logarithm. \(t\) is known as the heat kernel parameter and \(||.||_{F}\) is the Frobenius norm. As can be seen, the similarity between samples \(\mathbf{x}_{i}\) and \(\mathbf{x}_{l}\) is calculated based on the absolute distance \(||\mathbf{x}_{i}-\mathbf{x}_{l}||_{F}^{2}\). Approaches that rely on absolute distances face a common challenge in manifold space: samples may have close absolute distances while exhibiting significant actual differences, posing difficulties for accurate recognition. For instance, when considering two different faces both wearing the same scarf, distance-based algorithms may mistakenly recognize them as belonging to the same category. In order to leverage the discriminative information provided by the frequency domain, we innovatively proposed a kernel function for measuring the relationships between samples. Firstly, the similarity degree between frequency domain signal \(\mathbf{HF}_{i}\) and \(\mathbf{HF}_{l}\) corresponding to \(\mathbf{x}_{i}\) and \(\mathbf{x}_{l}\) is calculated according to their correlationship, denoted as:\[Cor(\mathbf{HF}_{i},\mathbf{HF}_{l})=\frac{vec(\mathbf{HF}_{i})^{T}vec(\mathbf{HF}_{ l})}{\sqrt{\|\mathbf{HF}_{i}\|_{F}^{2}\|\mathbf{HF}_{l}\|_{F}^{2}}}. \tag{7}\]  
Then, the similarity \(\mathbf{S}_{il}\) is calculated by:  
\[\mathbf{S}_{il}=exp(-\frac{t}{Cor(\mathbf{HF}_{i},\mathbf{HF}_{l})}). \tag{8}\]  
It is worth noting that, different from the absolute distance, the correlation coefficient between two signals will be larger when their similarity is higher. Hence, if \(\mathbf{x}_{i}\) and \(\mathbf{x}_{l}\) are more similar in the high-dimensional space, their discriminative frequency domain signals \(\mathbf{HF}_{i}\) and \(\mathbf{HF}_{l}\) should be closer to each other and the correlation \(Cor(\mathbf{HF}_{i},\mathbf{HF}_{l})\) should be larger, resulting in a greater similarity \(\mathbf{S}_{il}\). Conversely, when \(\mathbf{x}_{i}\) and \(\mathbf{x}_{l}\) are dissimilar, their similarity \(\mathbf{S}_{il}\) will be smaller. By substituting Equation (7) into (8), we can finally get the complete definition of the similarity measurement, named the high-frequency kernel function (HFK):  
\[\mathbf{S}_{il}=HFK(\mathbf{HF}_{i},\mathbf{HF}_{l})=exp(-\frac{t\sqrt{\| \mathbf{HF}_{i}\|_{F}^{2}\|\mathbf{HF}_{l}\|_{F}^{2}}}{vec(\mathbf{HF}_{i})^{ T}vec(\mathbf{HF}_{l})}). \tag{9}\]  
The high-frequency kernel function satisfies the definition and requirements of kernel functions, guaranteed by the _[PERSON]_ theorem and Proof 1. Proof. : **HFK satisfies the definition of the kernel function. ** based on the _[PERSON]_ theorem. The following two conditions need to be verified:  
The symmetry: for arbitrary pairwise input \(\mathbf{X}_{i}\) and \(\mathbf{X}_{j}\), HFK calculates the distance according to the correlation (7), which satisfies \(Cor(\mathbf{X}_{i},\mathbf{X}_{j})=Cor(\mathbf{X}_{j},\mathbf{X}_{i})\). Therefore, \(HKF(\mathbf{X}_{i},\mathbf{X}_{j})=HKF(\mathbf{X}_{j},\mathbf{X}_{i})\), proving that HFK is a symmetric function. The positive semi-definiteness: for any input \(\mathbf{X}_{i}\) and \(\mathbf{X}_{j}\), the Gram matrix of \(HFK(\mathbf{X}_{i},\mathbf{X}_{j})\) is a positive semi-definite matrix. Suppose matrix \(\mathbf{\hat{H}}\) is the Gram matrix of \(HFK(\mathbf{X}_{i},\mathbf{X}_{j})\):  
\[\mathbf{\hat{H}}=\left[