Approximation and sampling of multivariate probability distributions in the tensor train decomposition

TL;DR

This paper introduces a tensor-train based sampler for multivariate distributions that is efficient, scalable, and accurate, significantly outperforming traditional methods like DRAM in high-dimensional sampling tasks.

Contribution

The authors develop a novel tensor-train surrogate approach for sampling from complex multivariate distributions, enabling efficient high-dimensional sampling with linear storage scaling.

Findings

Tensor-train approximation enables efficient sampling with low storage costs.

The proposed methods outperform DRAM in all tested examples.

Importance-weighted quasi-Monte Carlo quadrature achieves the best efficiency.

Abstract

General multivariate distributions are notoriously expensive to sample from, particularly the high-dimensional posterior distributions in PDE-constrained inverse problems. This paper develops a sampler for arbitrary continuous multivariate distributions that is based on low-rank surrogates in the tensor-train format. We construct a tensor-train approximation to the target probability density function using the cross interpolation, which requires a small number of function evaluations. For sufficiently smooth distributions the storage required for the TT approximation is moderate, scaling linearly with dimension. The structure of the tensor-train surrogate allows efficient sampling by the conditional distribution method. Unbiased estimates may be calculated by correcting the transformed random seeds using a Metropolis--Hastings accept/reject step. Moreover, one can use a more efficient quasi-Monte Carlo quadrature that may be corrected either by a control-variate strategy, or by importance weighting. We show that the error in the tensor-train approximation propagates linearly into the Metropolis--Hastings rejection rate and the integrated autocorrelation time of the resulting Markov chain. These methods are demonstrated in three computed examples: fitting failure time of shock absorbers; a PDE-constrained inverse diffusion problem; and sampling from the Rosenbrock distribution. The delayed rejection adaptive Metropolis (DRAM) algorithm is used as a benchmark. We find that the importance-weight corrected quasi-Monte Carlo quadrature performs best in all computed examples, and is orders-of-magnitude more efficient than DRAM across a wide range of approximation accuracies and sample sizes. Indeed, all the methods developed here significantly outperform DRAM in all computed examples.