# Normalizing flow sampling with Langevin dynamics in the latent space

arXiv: https://arxiv.org/abs/2305.12149 · HTML: https://mlsanity.com/paper/2305.12149/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 50 references that are in the map.
- First posted on arXiv 20 May 2023.

## Builds on (most cited first, 24 of 24)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [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)
- [Do Deep Generative Models Know What They Don't Know?](https://mlsanity.com/paper/1810.09136/) (2018)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [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)
- [SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows](https://mlsanity.com/paper/2007.02731/) (2020)
- [Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Models](https://mlsanity.com/paper/2002.07101/) (2020)
- [Resampling Base Distributions of Normalizing Flows](https://mlsanity.com/paper/2110.15828/) (2021)
- [Stabilizing Invertible Neural Networks Using Mixture Models](https://mlsanity.com/paper/2009.02994/) (2020)
- [Langevin diffusions and the Metropolis-adjusted Langevin algorithm](https://mlsanity.com/paper/1309.2983/) (2013)
- [Local-Global MCMC kernels: the best of both worlds](https://mlsanity.com/paper/2111.02702/) (2021)
- [Variational Mixture of Normalizing Flows](https://mlsanity.com/paper/2009.00585/) (2020)
- [Sliced-Wasserstein normalizing flows: beyond maximum likelihood training](https://mlsanity.com/paper/2207.05468/) (2022)

## Built on by (most cited first, 7 of 7)

- [Outsourced diffusion sampling: Efficient posterior inference in latent spaces of generative models](https://mlsanity.com/paper/2502.06999/) (2025)
- [LATINO-PRO: LAtent consisTency INverse sOlver with PRompt Optimization](https://mlsanity.com/paper/2503.12615/) (2025)
- [Generative AI for Unmanned Vehicle Swarms: Challenges, Applications and Opportunities](https://mlsanity.com/paper/2402.18062/) (2024)
- [Learning few-step posterior samplers by unfolding and distillation of diffusion models](https://mlsanity.com/paper/2507.02686/) (2025)
- [Posterior Inference in Latent Space for Scalable Constrained Black-box Optimization](https://mlsanity.com/paper/2507.00480/) (2025)
- [Enhanced Importance Sampling through Latent Space Exploration in Normalizing Flows](https://mlsanity.com/paper/2501.03394/) (2025)
- [Bridging data-driven priors via the score function for posterior sampling -- Comparative review and experimental study](https://mlsanity.com/paper/2606.14800/) (2026)

## Abstract (arXiv)

Normalizing flows (NF) use a continuous generator to map a simple latent (e.g. Gaussian) distribution, towards an empirical target distribution associated with a training data set. Once trained by minimizing a variational objective, the learnt map provides an approximate generative model of the target distribution. Since standard NF implement differentiable maps, they may suffer from pathological behaviors when targeting complex distributions. For instance, such problems may appear for distributions on multi-component topologies or characterized by multiple modes with high probability regions separated by very unlikely areas. A typical symptom is the explosion of the Jacobian norm of the transformation in very low probability areas. This paper proposes to overcome this issue thanks to a new Markov chain Monte Carlo algorithm to sample from the target distribution in the latent domain before transporting it back to the target domain. The approach relies on a Metropolis adjusted Langevin algorithm (MALA) whose dynamics explicitly exploits the Jacobian of the transformation. Contrary to alternative approaches, the proposed strategy preserves the tractability of the likelihood and it does not require a specific training. Notably, it can be straightforwardly used with any pre-trained NF network, regardless of the architecture. Experiments conducted on synthetic and high-dimensional real data sets illustrate the efficiency of the method.