## 3 Preliminary
### Gaussian Process Latent Variable Model  
Let \(\mathbf{Y}=[\mathbf{y}_{1},\mathbf{y}_{2},\ldots,\mathbf{y}_{N}]^{\top}\in\mathbb{ R}^{N\times D}\) be \(D\)-dimensional observed variables with \(N\) samples. A GP-LVM assumes the existence of a projection from the \(Q\)-dimensional latent space to the \(D\)-dimensional observed space. Specifically, a GP-LVM assumes that the observed variables are generated from the Gaussian process (Kip  
where \(\
u\) and \(\kappa\) denote kernel parameters, \(\Gamma(\cdot)\) denotes the gamma function, and \(K_{\
u}(\cdot)\) denotes the modified Bessel function of the second kind. \(\
u\) is generally set to \(\frac{3}{2},\frac{5}{2}\), and they can be easily defined as follows:  
\[k_{Mat\acute{e}rn\,3/2}(\mathbf{x},\mathbf{x}^{\prime})=\sigma^{2 }\Bigg{(}1+\frac{\sqrt{3}||\mathbf{x}-\mathbf{x}^{\prime}||}{\kappa}\Bigg{)} \exp\Bigg{(}-\frac{\sqrt{3}||\mathbf{x}-\mathbf{x}^{\prime}||}{\kappa}\Bigg{)}, \tag{8}\] \[k_{Mat\acute{e}rn\,5/2}(\mathbf{x},\mathbf{x}^{\prime})=\sigma^{2 }\Bigg{(}1+\frac{\sqrt{5}||\mathbf{x}-\mathbf{x}^{\prime}||}{\kappa}+\frac{5 ||\mathbf{x}-\mathbf{x}^{\prime}||^{2}}{3\kappa^{2}}\Bigg{)}\exp\Bigg{(}- \frac{\sqrt{5}||\mathbf{x}-\mathbf{x}^{\prime}||}{\kappa}\Bigg{)} \tag{9}\]  
Furthermore, if \(\
u\rightarrow\infty\), the Matern kernel converges to the SE kernel in Equation (6). Although the SE and Matern kernels have different expressions, they are both homogeneous kernels as \(k(\mathbf{x},\mathbf{x}^{\prime})=k(||\mathbf{x}-\mathbf{x}^{\prime}||)\). The stationary kernel is in a broader class of homogeneous kernels as \(k(\mathbf{x},\mathbf{x}^{\prime})=k(\mathbf{x}-\mathbf{x}^{\prime})\). The SM kernel [30] is a stationary kernel and can measure the periodicity of input signals. The SM kernel is defined by the following equation:  
\[k_{SM}(\mathbf{x},\mathbf{x}^{\prime})=\sum_{p=1}^{N_{w}}w_{p}\prod_{q=1}^{Q} \exp\left[-2\pi^{2}(x_{q}-x_{q}^{\prime})^{2}v_{q}^{(p)}\right]\cos\left[2\pi (x_{q}-x_{q}^{\prime})\mu_{q}^{(p)}\right], \tag{10}\]  
where \(N_{w}\) denotes the number of mixtures, \(w_{p}\) denotes a weight parameter of each mixture, \(v_{q}^{(p)}\) denotes a lengthscale parameter of the dimension \(q\) of a mixture \(p\), and \(\mu_{q}^{(p)}\) denotes a mean parameter of the dimension \(q\) of a mixture \(p\). We collectively denote \(\boldsymbol{\mu}_{p}=[\mu_{1}^{(p)},\mu_{2}^{(p)},\ldots,\mu_{Q}^{(p)}]^{\top} \in\mathbb{R}^{Q}\). The SM kernel is derived from [PERSON]'s theorem, which determines the spectral density of any stationary kernel. On this basis, \(\mu_{q}^{(p)}\) indicates the center in the spectral domain, and the SE kernel is the special case of the SM kernel as \(\mu_{q}^{(p)}=0\). The SM kernel measures the similarity between two inputs with the distance \(||\mathbf{x}-\mathbf{x}^{\prime}||\) and the periodicity represented by the cosine function of the spectral \(\boldsymbol{\mu}_{p}\). By optimizing these parameters, we can find the optimal center in the spectral domain from the input signals and derive a data-specific expression of the kernel. For intuitive comparison of these kernels, we present the sampling results from \(\mathcal{GP}(\mathbf{0},k(\cdot,\cdot))\) for each kernel in Figure 1.  
Figure 1: Sampling results from \(\mathcal{GP}(0,k(\cdot,\cdot))\) for each kernel function. Since the SE kernel is a special case of the SM kernel, they produce similar results.