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Low-rank tensor approximations have shown great potential for uncertaintyquantification in high dimensions, for example, to build surrogate models thatcan be used to speed up large-scale inference problems (Eigel et al., InverseProblems 34, 2018; Dolgov et al., Statistics & Computing 30, 2020). Thefeasibility and efficiency of such approaches depends critically on the rankthat is necessary to represent or approximate the underlying distribution. Inthis paper, a-priori rank bounds for approximations in the functionaltensor-train representation for the case of Gaussian models are developed. Itis shown that under suitable conditions on the precision matrix, the Gaussiandensity can be approximated to high accuracy without suffering from anexponential growth of complexity as the dimension increases. These resultsprovide a rigorous justification of the suitability and the limitations oflow-rank tensor methods in a simple but important model case. Numericalexperiments confirm that the rank bounds capture the qualitative behavior ofthe rank structure when varying the parameters of the precision matrix and theaccuracy of the approximation. Finally, the practical relevance of thetheoretical results is demonstrated in the context of a Bayesian filteringproblem.

Rank Bounds for Approximating Gaussian Densities in the Tensor-Train Format