Transport Quasi-Monte Carlo

TL;DR

This paper introduces a transport map-based Quasi-Monte Carlo method that improves high-dimensional integral estimation accuracy, especially for complex distributions, by leveraging invertible transformations inspired by normalizing flows.

Contribution

The paper proposes a novel transport QMC approach using flexible invertible maps to extend QMC applicability to complex, unnormalized target distributions, with theoretical convergence guarantees.

Findings

Achieves faster convergence rates than standard Monte Carlo.

Effective in Bayesian inference tasks with complex distributions.

Theoretical analysis confirms improved error bounds.

Abstract

Quasi-Monte Carlo (QMC) is a powerful method for evaluating high-dimensional integrals. However, its use is typically limited to distributions where direct sampling is straightforward, such as the uniform distribution on the unit hypercube or the Gaussian distribution. For general target distributions with potentially unnormalized densities, leveraging the low-discrepancy property of QMC to improve accuracy remains challenging. We propose training a transport map to push forward the uniform distribution on the unit hypercube to approximate the target distribution. Inspired by normalizing flows, the transport map is constructed as a composition of simple, invertible transformations. To ensure that RQMC achieves its superior error rate, the transport map must satisfy specific regularity conditions. We introduce a flexible parametrization for the transport map that not only meets these conditions but is also expressive enough to model complex distributions. Our theoretical analysis establishes that the proposed transport QMC estimator achieves faster convergence rates than standard Monte Carlo, under mild and easily verifiable growth conditions on the integrand. Numerical experiments confirm the theoretical results, demonstrating the effectiveness of the proposed method in Bayesian inference tasks.