# Kernel Learning by Spectral Representation and Gaussian Mixtures
## 2 The Concept of Kernels
### Locally Stationary Kernels  
Extending on the previous concept, [PERSON] [22] defines the locally stationary kernels as:  
\[\kappa(\textbf{x},\textbf{x}):=\kappa_{1}\bigg{(}\frac{\textbf{x}+\textbf{x}^{ \prime}}{2}\bigg{)}\kappa_{s}(\textbf{x}-\textbf{x}^{\prime}), \tag{2}\]  
where \(\kappa_{1}\) is a non negative function and \(\kappa_{s}\) is a stationary kernel. This type of kernels increase the power of the representation by introducing a possible variance into the final calculated similarity through the use of \(\kappa_{1}\). Furthermore, we can see from Equation (2) that the Locally Stationary Kernels include all stationary kernels. In order to see this, we make \(\kappa_{1}(\cdot)=c\), where \(c\) is a positive constant, then \(\kappa(\textbf{x},\textbf{x}):=c\kappa_{s}(\textbf{x}-\textbf{x}^{\prime})\), a multiple of all stationary kernels. Furthermore, the variance of locally stationary kernels is given by \(\textbf{x}=\textbf{x}^{\prime}\), thus, the variance is defined as:  
\[\kappa(\textbf{x},\textbf{x})=\kappa(\textbf{x})\kappa(\textbf{0})=\kappa_{1} (\textbf{x}),\]  
This means that the variance of the Locally Stationary Kernels relies in the positive definite function \(\kappa_{1}\).  
The spectral representation of a locally stationary kernel is also given by [22], and it is defined as:  
\[\kappa_{1}\bigg{(}\frac{\textbf{x}+\textbf{x}^{\prime}}{2}\bigg{)}\kappa_{s} (\textbf{x}-\textbf{x}^{\prime})=\int_{\mathcal{X}}\int_{\mathcal{X}}exp \bigg{(}i\frac{\mathbf{\omega}_{1}^{T}\textbf{x}-\mathbf{\omega}_{2}^{T}\textbf{x}^{ \prime}}{2}\bigg{)}f_{1}\bigg{(}\frac{\mathbf{\omega}_{1}+\mathbf{\omega}_{2}}{2} \bigg{)}f_{2}(\mathbf{\omega}_{1}-\mathbf{\omega}_{2})d\mathbf{\omega}_{1}d\mathbf{\omega}_{2}.\]  
Furthermore, by setting \(\textbf{x}=\textbf{x}^{\prime}=\textbf{0}\), we can get:  
\[\kappa(\textbf{0},\textbf{0})=\int_{\mathcal{X}}\int_{\mathcal{X}}f_{1}\bigg{(} \frac{\mathbf{\omega}_{1}+\mathbf{\omega}_{2}}{2}\bigg{)}f_{2}(\mathbf{\omega}_{1}-\mathbf{ \omega}_{2})d\mathbf{\omega}_{1}d\mathbf{\omega}_{2}.\]  
Consequently, in order to define a locally stationary kernel, \(f_{1}\) and \(f_{2}\) must be integrable functions. Additionally, an important fact is that the kernel \(\kappa\) has a defined inverse, given by:  
\[f_{1}\bigg{(}\frac{\mathbf{\omega}_{1}+\mathbf{\omega}_{2}}{2}\bigg{)}f_{2}(\mathbf{\omega }_{1}-\mathbf{\omega}_{2})=\frac{1}{(2\pi)^{2}}\int_{\mathcal{X}}\int_{\mathcal{X }}\kappa_{1}\bigg{(}\frac{\textbf{x}_{1}+\textbf{x}^{\prime}}{2}\bigg{)}\kappa _{2}(\textbf{x}-\textbf{x}^{\prime})d\textbf{x}d\textbf{x}^{\prime}.\]  
Moreover, \(f_{2}\) is the Fourier transform of \(\kappa_{1}\) and \(f_{1}\) is the Fourier transform of \(\kappa_{2}\). Thus, if we introduce two dummy variables \(\textbf{u}=(\textbf{x}+\textbf{x}^{\prime})/2\) and \(\textbf{v}=\textbf{x}-\textbf{x}^{\prime}\), it is possible to obtain:  
\[f_{1}(\mathbf{\omega}_{1})=\frac{1}{2\pi}\int_{\mathcal{X}}exp(-i \textbf{v}^{T}\mathbf{\omega}_{1})\kappa_{2}(\textbf{v})d\mathbf{v}\] \[f_{2}(\mathbf{\omega}_{2})=\frac{1}{2\pi}\int_{\mathcal{X}}exp(-i \textbf{u}^{T}\mathbf{\omega}_{2})\kappa_{1}(\textbf{u})d\textbf{u}\]  
and  
\[\kappa_{1}(\textbf{u})=\int_{\mathcal{X}}exp(i\textbf{u}^{T}\mathbf{ \omega}_{2})f_{2}(\mathbf{\omega}_{2})d\mathbf{\omega}_{2}\] \[\kappa_{2}(\textbf{v})=\int_{\mathcal{X}}exp(i\textbf{v}^{T}\mathbf{ \omega}_{1})f_{1}(\mathbf{\omega}_{1})d\mathbf{\omega}_{1},\]  
with this in mind, it is possible to use the ideas in [16] to approximate the locally stationary kernels.