Kernel Trick – Is $\min(k_1(x, y), k_2(x, y))$ a Positive Definite Kernel?

Question

It is known that $(x,y)\in \mathbb{R}^2 \mapsto \min(x,y)$ is a positive definite kernel. Can we generalize this result in the following way :
Let $k_1(x, y)$ and $k_2(x, y)$ be any two positive definite kernels. Is $$k:(x,y)\in X^2 \mapsto \min(k_1(x, y), k_2(x, y))$$ a positive definite kernel?

I think it is a psd. I tried proving it using the definition of psds but it seems it is not easy. And help would be appreciated.

Answer 1

Let the following be two kernel matrices: $$K_1=\begin{bmatrix}4 & -2\\-2&4\end{bmatrix}, K_2=\begin{bmatrix}1 & 0.5\\0.5&1\end{bmatrix}$$

$$K=\min(K_1,K_2)=\begin{bmatrix}1&-2\\-2&1\end{bmatrix}$$

But, $K$ is not PSD or PD.