Approximation and sampling of multivariate probability distributions in the tensor train decomposition
Abstract
General multivariate distributions are notoriously expensive to sample from, particularly the highdimensional posterior distributions in PDEconstrained inverse problems. This paper develops a sampler for arbitrary continuous multivariate distributions that is based on lowrank surrogates in the tensortrain format. We construct a tensortrain 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 tensortrain surrogate allows efficient sampling by the conditional distribution method. Unbiased estimates may be calculated by correcting the transformed random seeds using a MetropolisHastings accept/reject step. Moreover, one can use a more efficient quasiMonte Carlo quadrature that may be corrected either by a controlvariate strategy, or by importance weighting. We show that the error in the tensortrain approximation propagates linearly into the MetropolisHastings 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 PDEconstrained 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 importanceweight corrected quasiMonte Carlo quadrature performs best in all computed examples, and is ordersofmagnitude 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 DOI:
 10.48550/arXiv.1810.01212
 arXiv:
 arXiv:1810.01212
 Bibcode:
 2018arXiv181001212D
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Probability;
 Mathematics  Statistics Theory;
 65D15;
 65D32;
 65C05;
 65C40;
 65C60;
 62F15;
 15A69;
 15A23
 EPrint:
 32 pages