  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. These results
provide a rigorous justification of the suitability and the limitations of
low-rank tensor methods in a simple but important model case. Numerical
experiments confirm that the rank bounds capture the qualitative behavior of
the rank structure when varying the parameters of the precision matrix and the
accuracy of the approximation. Finally, the practical relevance of the
theoretical results is demonstrated in the context of a Bayesian filtering
problem.
