From Euclidean geometry we know that a matrix $C$ is a matrix of squared Euclidean distances between some points if and only if $-\frac{1}{2} H D H \succeq 0$ (positive semi-definite) with $H = (I - \frac{1}{n} 1 1^\top)$ being the centering matrix [1].
I am wondering if there are similar results for the Gaussian kernel between some points, that is a characterization showing that a matrix $C$ can be written as $C = (\exp(-\|x_i-x_j\|_2^2)$ for some point $x_1, \cdots, x_n$.
Obviously one simple characterization would be $\frac{1}{2} H \log C H \succeq 0$ with the logarithm taken pointwise but I am wondering if there is a ‘‘nicer'' characterization.
[1] J. C. Gower, “Euclidean Distance Geometry,” Math. Sci., vol. 7, pp. 1–14, 1982.
