Deep importance sampling using tensor trains with application to a priori and a posteriori rare event estimation

TL;DR

This paper introduces a deep importance sampling approach using tensor trains to efficiently estimate rare event probabilities in high-dimensional Bayesian problems, overcoming computational challenges associated with concentrated densities.

Contribution

It develops a scalable tensor-train based method for approximating optimal importance distributions and normalizing constants, improving variance reduction in rare event probability estimation.

Findings

Efficient estimation of rare event probabilities in high dimensions.

Little to no increase in computational complexity as event probability decreases.

Able to estimate previously unattainable rare event probabilities in complex models.

Abstract

We propose a deep importance sampling method that is suitable for estimating rare event probabilities in high-dimensional problems. We approximate the optimal importance distribution in a general importance sampling problem as the pushforward of a reference distribution under a composition of order-preserving transformations, in which each transformation is formed by a squared tensor-train decomposition. The squared tensor-train decomposition provides a scalable ansatz for building order-preserving high-dimensional transformations via density approximations. The use of composition of maps moving along a sequence of bridging densities alleviates the difficulty of directly approximating concentrated density functions. To compute expectations over unnormalized probability distributions, we design a ratio estimator that estimates the normalizing constant using a separate importance distribution, again constructed via a composition of transformations in tensor-train format. This offers better theoretical variance reduction compared with self-normalized importance sampling, and thus opens the door to efficient computation of rare event probabilities in Bayesian inference problems. Numerical experiments on problems constrained by differential equations show little to no increase in the computational complexity with the event probability going to zero, and allow to compute hitherto unattainable estimates of rare event probabilities for complex, high-dimensional posterior densities.