prove If $k_1$ and $k_2$ are positive semidefinite kernels then $min\{k_1, k_2\}$ and $max\{k_1, k_2\}$ are psd too.

Question

I can prove for $R+$ the function $min(x,y)$ is a positive semidefinite kernel. But I'm stuck in proving the following statement.

Suppose $k_1(x,y)$ and $k_2(x,y)$ are positive semidefinite kernels from $\chi \times \chi \rightarrow R $

Prove both $k_{min}$ and $k_{max}$ are positive semidefinite kernels too

• $k_{min}(x,y) = min\{k_1(x,y), k_2(x,y)\}$
• $k_{max}(x,y) = max\{k_1(x,y), k_2(x,y)\}$

Answer 1

The conclusion is not true for $\max$ (see). The conclusion fails also for $\min.$ Let $$ K_1=\begin{pmatrix}1 & 2\\ 2& 5 \end{pmatrix},\qquad K_2=\begin{pmatrix}5 & 2\\ 2& 1 \end{pmatrix}$$ be the matrices corresponding to $k_1$ and $k_2.$ Then $\min(k_1,k_2) $ corresponds to the matrix $$ \begin{pmatrix}1 & 2\\ 2& 1 \end{pmatrix}$$ which is not positive definite.