核函数与希尔伯特空间

A kernel function$\kappa(\cdot,\cdot)$ is defined on a Reproducing Kernel Hilbert Space (RKHS)$\mathcal{H}$, where it serves as a similarity measure between feature vectors$\mathbf{x}_i, \mathbf{x}_j\in \mathcal{X}$. The kernel satisfies:

$$\begin{equation*} \kappa(\mathbf x_i, \mathbf x_j)=\langle\phi(\mathbf x_i), \phi(\mathbf x_j)\rangle_{\mathcal H}=\phi(\mathbf x_i)^\top\phi(\mathbf x_j), \end{equation*}$$

where$\phi(\cdot):\mathcal{X}\mapsto\mathcal{H}$ is an implicit feature mapping that embeds the input feature space into RKHS. This formulation ensures that$\kappa(\cdot,\cdot)$ is symmetric and positive semi-definite, aligning with Mercer’s theorem and enabling kernel methods for spatial analysis.

参考

1. https://www.cnblogs.com/luzhanshi/articles/18895084
2. https://www.cnblogs.com/massquantity/p/11110397.html
3. https://zhuanlan.zhihu.com/p/441182447
4. https://zhuanlan.zhihu.com/p/541226732
5. https://www.cnblogs.com/zhangcn/p/13289236.html