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

TL;DR

This paper establishes theoretical bounds on the tensor-train ranks needed to accurately approximate Gaussian densities, demonstrating that low-rank tensor methods are feasible in high-dimensional settings under certain conditions.

Contribution

It provides the first a-priori rank bounds for Gaussian densities in the tensor-train format, clarifying when low-rank approximations are effective in high dimensions.

Findings

Rank bounds accurately predict the complexity of Gaussian approximations.

Numerical experiments validate the theoretical bounds.

Application to Bayesian filtering demonstrates practical relevance.

Abstract

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.