Weighted Minkowski RBF kernel

Question

The radial basis function (RBF) kernel is given by

$$K_{\text{RBF}}(\mathbf{x}, \mathbf{y})=\exp[-\gamma\|\mathbf{x}-\mathbf{y}\|^2_2]$$

where $\|\mathbf{x}-\mathbf{y}\|^2_2$ is the squared Euclidean distance. I have three questions. The answer to all three may be generalized RBF kernel, but I would like to make sure.

First, what is the name of the kernel that results if the Euclidean distance is not squared?

$$K(\mathbf{x}, \mathbf{y})=\exp[-\gamma\|\mathbf{x}-\mathbf{y}\|_2]$$

Second, what is the name of the kernel that results if another Minkowski $p$-norm is taken and raised to the $n^{th}$ power?

$$K_{p,n}(\mathbf{x}, \mathbf{y})=\exp[-\gamma\|\mathbf{x}-\mathbf{y}\|_p^n]$$ $$K_{2,2}=K_{\text{RBF}}$$

And finally, what is the name of the kernel that results if that Minkowski $p$-norm is weighted?

$$K_{p,n, \mathbf{w}}(\mathbf{x}, \mathbf{y})=\exp\left[-\gamma\left(\sqrt[p]{\sum_iw_i|x_i-y_i|^p}\right)^n\right]$$

Answer 1

The first one is called (by some) "exponential covariance function". Covariance relates to the interpretation of the RBF interpolation procedure as a regression task. You can read more atthis book (specifically - the chapter that discusses covariance functions).