Low-Rank Variance Approximation in GMRF Models: Single and Multiscale Approaches

We present a versatile framework for tractable computation of approximate variances in large-scale Gaussian Markov random field estimation problems. In addition to its efficiency and simplicity, it also provides accuracy guarantees. Our approach relies on the construction of a certain low-rank aliasing matrix with respect to the Markov graph of the model. We first construct this matrix for single-scale models with short-range correlations and then introduce spliced wavelets and propose a construction for the long-range correlation case, and also for multiscale models. We describe the accuracy guarantees that the approach provides and apply the method to a large interpolation problem from oceanography with sparse, irregular, and noisy measurements, and to a gravity inversion problem.