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

arXiv (Cornell University)

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.

Authors 2

1. Lars Grasedyck Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik , RWTH Aachen , Templergraben 55 , 52056 Aachen , Germany

2. Robert Scheichl

   Heidelberg University · University of Bath

   Affiliation as printed

   Department of Mathematical Sciences , University of Bath , Claverton Down , Bath BA2 7AY , United Kingdom

   Institute for Applied Mathematics , Heidelberg University , Im Neuenheimer Feld 205 , 69120 Heidelberg , Germany