# On Sampling with Approximate Transport Maps

arXiv: https://arxiv.org/abs/2302.04763 · HTML: https://mlsanity.com/paper/2302.04763/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 83 references that are in the map.
- First posted on arXiv 9 Feb 2023.

## Builds on (most cited first, 30 of 34)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Semi-Supervised Learning with Normalizing Flows](https://mlsanity.com/paper/1912.13025/) (2019)
- [NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport](https://mlsanity.com/paper/1903.03704/) (2019)
- [Your GAN is Secretly an Energy-based Model and You Should use Discriminator Driven Latent Sampling](https://mlsanity.com/paper/2003.06060/) (2020)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm](https://mlsanity.com/paper/2012.12810/) (2020)
- [Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients](https://mlsanity.com/paper/1905.12247/) (2019)
- [Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling](https://mlsanity.com/paper/2109.13055/) (2021)
- [Gauge-equivariant flow models for sampling in lattice field theories with pseudofermions](https://mlsanity.com/paper/2207.08945/) (2022)
- [Variational Refinement for Importance Sampling Using the Forward Kullback-Leibler Divergence](https://mlsanity.com/paper/2106.15980/) (2021)
- [MCMC Should Mix: Learning Energy-Based Model with Neural Transport Latent Space MCMC](https://mlsanity.com/paper/2006.06897/) (2020)
- [Aspects of scaling and scalability for flow-based sampling of lattice QCD](https://mlsanity.com/paper/2211.07541/) (2022)
- [Flow-based sampling for multimodal and extended-mode distributions in lattice field theory](https://mlsanity.com/paper/2107.00734/) (2021)
- [Local-Global MCMC kernels: the best of both worlds](https://mlsanity.com/paper/2111.02702/) (2021)
- [Transport Elliptical Slice Sampling](https://mlsanity.com/paper/2210.10644/) (2022)
- [Machine Learning of Thermodynamic Observables in the Presence of Mode Collapse](https://mlsanity.com/paper/2111.11303/) (2021)
- [Deterministic Langevin Monte Carlo with Normalizing Flows for Bayesian Inference](https://mlsanity.com/paper/2205.14240/) (2022)
- [Convergence of Gibbs Sampling: Coordinate Hit-and-Run Mixes Fast](https://mlsanity.com/paper/2009.11338/) (2020)

## Built on by (most cited first, 21 of 21)

- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Stochastic Localization via Iterative Posterior Sampling](https://mlsanity.com/paper/2402.10758/) (2024)
- [Markovian Flow Matching: Accelerating MCMC with Continuous Normalizing Flows](https://mlsanity.com/paper/2405.14392/) (2024)
- [Learned Reference-based Diffusion Sampling for multi-modal distributions](https://mlsanity.com/paper/2410.19449/) (2024)
- [Improving the evaluation of samplers on multi-modal targets](https://mlsanity.com/paper/2504.08916/) (2025)
- [Balanced Training of Energy-Based Models with Adaptive Flow Sampling](https://mlsanity.com/paper/2306.00684/) (2023)
- [Self-Tuning Hamiltonian Monte Carlo for Accelerated Sampling](https://mlsanity.com/paper/2309.13593/) (2023)
- [Amortized Bayesian Workflow](https://mlsanity.com/paper/2409.04332/) (2024)
- [Combining Normalizing Flows and Quasi-Monte Carlo](https://mlsanity.com/paper/2401.05934/) (2024)
- [Parallel Affine Transformation Tuning of Markov Chain Monte Carlo](https://mlsanity.com/paper/2401.16567/) (2024)
- [Convergence Rates for Distribution Matching with Sliced Optimal Transport](https://mlsanity.com/paper/2602.10691/) (2026)
- [Sequential transport maps using SoS density estimation and α-divergences](https://mlsanity.com/paper/2402.17943/) (2024)
- [Fast Bayesian Inference for Neutrino Non-Standard Interactions at Dark Matter Direct Detection Experiments](https://mlsanity.com/paper/2405.14932/) (2024)
- [Empirical evaluation of normalizing flows in Markov Chain Monte Carlo](https://mlsanity.com/paper/2412.17136/) (2024)
- [Sampling by averaging: A multiscale approach to score estimation](https://mlsanity.com/paper/2508.15069/) (2025)
- [Latent-IMH: Efficient Bayesian Inference for Inverse Problems with Approximate Operators](https://mlsanity.com/paper/2601.20888/) (2026)
- [MCMC-driven learning](https://mlsanity.com/paper/2402.09598/) (2024)
- [Reducing normalizing flow complexity for MCMC preconditioning](https://mlsanity.com/paper/2511.02345/) (2025)
- [Entropic Mirror Monte Carlo](https://mlsanity.com/paper/2602.03165/) (2026)
- [Neural Transport Nested Sampling](https://mlsanity.com/paper/2609.29413/) (2026)
- [Flow Matching for Fast Posterior Sampling in Bayesian Inverse Problems](https://mlsanity.com/paper/2610.02377/) (2026)

## Abstract (arXiv)

Transport maps can ease the sampling of distributions with non-trivial geometries by transforming them into distributions that are easier to handle. The potential of this approach has risen with the development of Normalizing Flows (NF) which are maps parameterized with deep neural networks trained to push a reference distribution towards a target. NF-enhanced samplers recently proposed blend (Markov chain) Monte Carlo methods with either (i) proposal draws from the flow or (ii) a flow-based reparametrization. In both cases, the quality of the learned transport conditions performance. The present work clarifies for the first time the relative strengths and weaknesses of these two approaches. Our study concludes that multimodal targets can be reliably handled with flow-based proposals up to moderately high dimensions. In contrast, methods relying on reparametrization struggle with multimodality but are more robust otherwise in high-dimensional settings and under poor training. To further illustrate the influence of target-proposal adequacy, we also derive a new quantitative bound for the mixing time of the Independent Metropolis-Hastings sampler.