How to derive an inverse of Gaussian Kernel As an example, say I have a function (Gaussian process kernel):
$$K(x_i, x_j) \equiv \exp(-\alpha |x_i-x_j|^2)+\beta \delta_{ij}$$
Is there a way to analytically express $K^{-1}(x_i,x_j)$, s.t. matrices $K$ and $K^{-1}$ made from $K(x_i,x_j)$ and $K^{-1}(x_i,x_j)$ are inverses of each other? An approximation would also be great.
A similar question has been asked before (link), but I think that is for numerical calculation of $K^{-1}$. I am seeking an analytic expression or formula for $K^{-1}(x_i,x_j)$.

This would be hard, at least for these commonly used kernels like Matern, squared exponential etc.
Suppose that you can obtain an analytical solution of your $K^{-1}(x_i, x_i)$. Then GP regression problem can be solve in $O(N^2)$ time, as there is no more matrix inversion but simply matrix multiplications. This would be a great significance for GP, as you have reduced the complexity $O(N^3)$ to $O(N^2)$ without any approximation. Although I cannot give any math proof to show that this is hard, it would have been done decades ago if it is easy.
