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

Low rank tensor approximations have been employed successfully, for example, to build surrogate models that can be used to speed up large-scale inference problems in high dimensions. The success of this depends critically on the rank that is necessary to represent or approximate the underlying distribution. In this paper, we develop a-priori rank bounds for approximations in the functional Tensor-Train representation for the case of a Gaussian (normally distributed) model. We show that under suitable conditions on the precision matrix, we can represent the Gaussian density to high accuracy without suffering from an exponential growth of complexity as the dimension increases. Our results provide evidence of the suitability and 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.