Let a gaussian kernel be defined as $K(x_i, x_j) \equiv \exp(-\alpha |x_i-x_j|^2)$ and define the kernel matrix of some set of datapoints $\{x_i\}_{i=1}^n$ which are regularly spaced in 2D lexicographically, as the $n\times n$ matrix $K$ with $K_{ij}=K(x_i,x_j)$, Suppose that I make $K$ sparse somehow that each row has 9 non-zero entries at most. For example by filling $K_{ij}$ only around a $3\times 3$ neighborhood for a fixed $x_i$ and fill other elements of that row with zero.
Given that in this case $K$ is sparse and symmetric, I am looking for an analytical representation for $K^{-1}(x_i,x_j)$ in terms of elements of $K$.
I did a search and find some similar questions:
Inverse of Gaussian Kernel Matrix
How to derive an inverse of Gaussian Kernel
https://stats.stackexchange.com/questions/549075/
But none of them could help me. I think that I can leverage of sparsity of $K$ in my case as no matter how large $K$ is, only 9 entries are non-zero at each row at most, I tried to take a look at methods for sparse inverse covariance matrix approximation but the problem is that I am looking for an analytical representation for $K^{-1}$ not a numerical solution. Even an approximation for $K^{-1}$ would be appreciated.