Mercer Theorem

#Computers #Math

$$

\boldsymbol{K}=\begin{pmatrix}
k(x_{1},x_{1}) & \ldots & k(x_{1},x_{N}) \\\vdots & \ddots & \vdots \\k(x_{N},x_{1}) & \ldots & k(x_{N},x_{N})
\end{pmatrix}
$$

• $\displaystyle k(x_{n},x_{m})$ is a kernel function iff $\displaystyle ~\forall~\text{finite }N\text{ and }x_{1},\ldots, x_{N}$, this matrix is a positive semidefinite matrix