Limitations on low rank approximations for covariance matrices of spatial data

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Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets (2014, unknown): “Computation issues Needs to store the n2 pairwise distances to compute C(θ) C(θ) is dense, computing C(θ)−1 uses O(n3) flops Computationally infeasible for such massive datasets Low rank models Regresses w on w̃ realized at r locations (knots) where r n O(nr2) flops but requires large r for massive ”
Likelihood approximation with hierarchical matrices for large spatial datasets (2017, background): “…sometimes performs even worse than assuming independent blocks in the covariance (Stein, 2013); low-rank approximations have their own limitations (Stein, 2014); andMarkovmodels depend on the observation locations, requiring irregular locations to be realigned on a much finer grid with missing…”
A General Framework for Vecchia Approximations of Gaussian Processes (2017, background): “When each subvector or block corresponds to a contiguous subregion in space, Stein (2014) showed that this approximation can be quite competitive as a surrogate for the likelihood, and also computationally inexpensive since each term in the product incurs a small computational cost and can be…”