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

Understand

Low-rank tensor approximations have shown great potential for uncertainty quantification in high dimensions, for example, to build surrogate models that can be used to speed up large-scale inference problems (Eigel et al., Inverse Problems 34, 2018; Dolgov et al., Statistics & Computing 30, 2020).

• The feasibility and efficiency of such approaches depends critically on the rank that is necessary to represent or approximate the underlying distribution.
• In this paper, a-priori rank bounds for approximations in the functional tensor-train representation for the case of Gaussian models are developed.
• It is shown that under suitable conditions on the precision matrix, the Gaussian density can be approximated to high accuracy without suffering from an exponential growth of complexity as the dimension increases.