TITLE: Inverse of Gaussian Kernel Matrix
QUESTION [6 upvotes]: Let a gaussian kernel be defined as $K(x_i, x_j) \equiv \exp(-\alpha |x_i-x_j|^2)+\beta \delta_{ij}$, and define the kernel matrix of some set of datapoints $\{x_i\}_{i=1}^n$ as the $n\times n$ matrix $K$ with $K_{ij} = K(x_i, x_j)$.
This is a common construction in various fields, e.g. Gaussian Processes. Is there a fast way of calculating the inverse of the kernel matrix?

Thoughts: if we could break down the matrix $K$ into the form $\beta I + u u^T$ for some column vector $u$, we could use the Sherman–Morrison formula to quickly calculate the inverse, however, we know such a $u$ would be infinite dimensional. Is there another trick one could use?

REPLY [3 votes]: Much works has been done on the scalability of Gaussian Processes. The main idea is to approximate your kernel matrix $K$ with another matrix that admits faster operations.  This allows us to have fast matrix vector multiplications when optimizing hyper-parameters and use iterative algorithms such as Conjugate Gradients or GMRES to invert $K$.
The Structured Kernel Interpolation(SKI) paper gives a method for approximating your data with data on a grid, which allows you to use inducing point methods, toeplitz, kronecker methods to get the fast computations. See https://arxiv.org/abs/1503.01057