# Transport Quasi-Monte Carlo

arXiv: https://arxiv.org/abs/2412.16416 · HTML: https://mlsanity.com/paper/2412.16416/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 21 Dec 2024.

## Builds on (most cited first, 9 of 9)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Certified dimension reduction in nonlinear Bayesian inverse problems](https://mlsanity.com/paper/1807.03712/) (2018)
- [Quasi-Monte Carlo Variational Inference](https://mlsanity.com/paper/1807.01604/) (2018)
- [Variational Gaussian Copula Inference](https://mlsanity.com/paper/1506.05860/) (2015)
- [Transporting Higher-Order Quadrature Rules: Quasi-Monte Carlo Points and Sparse Grids for Mixture Distributions](https://mlsanity.com/paper/2308.10081/) (2023)
- [Langevin Quasi-Monte Carlo](https://mlsanity.com/paper/2309.12664/) (2023)
- [Combining Normalizing Flows and Quasi-Monte Carlo](https://mlsanity.com/paper/2401.05934/) (2024)
- [Conditional Quasi-Monte Carlo with Constrained Active Subspaces](https://mlsanity.com/paper/2212.13232/) (2022)

## Built on by (most cited first, 2 of 2)

- [A friendly introduction to triangular transport](https://mlsanity.com/paper/2503.21673/) (2025)
- [Rotated Mean-Field Variational Inference and Iterative Gaussianization](https://mlsanity.com/paper/2510.07732/) (2025)

## Abstract (arXiv)

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 QMC 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.