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How to prove $k(x_i,x_j)=e^{-(LR(x_i-x_j))^TLR(x_i-x_j)}$ is a valid kernel function or positive semi definite?

$x=(\mu,\lambda)^T$ and R is a 2x2 rotation matrix, L is a 2x2 diagonal scaling matrix with positive entries. Any idea is appreciated.

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One very routine proof is to employ Schoenberg's interpolation theorem (see chapter 15 of A Course in Approximation Theory by Cheney and Light or sec. 2.5 of this book chapter, for instances).

The radial basis function $\phi(y) = \exp(-y)$ is completely monotone on $[0,\infty)$. By Schoenberg's interpolation theorem , $\phi(\|\cdot\|^2)$ is strictly positive definite. Therefore $f(u,v)=\phi(\|u-v\|^2)=e^{-\|u-v\|^2}$ is a kernel. As $LR$ is invertible, it follows that the matrix $\left(k(x_i,x_j)\right)_{i,j=1,2,\ldots,n}=\left(f(LRx_i,LRx_j)\right)_{i,j=1,2,\ldots,n}$ is positive definite if all the $x_i$s are distinct, or positive semidefinite otherwise.

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  • $\begingroup$ How to derive this step $(k(x_i,x_j))_{i,j=1,2,…,n}=(f(LRx_i,LRx_j))_{i,j=1,2,…,n}$ ? $\endgroup$ – user40577 Feb 27 '14 at 3:36

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