# MCMC Should Mix: Learning Energy-Based Model with Neural Transport Latent Space MCMC

arXiv: https://arxiv.org/abs/2006.06897 · HTML: https://mlsanity.com/paper/2006.06897/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2020.

## Built on by (most cited first, 26 of 26)

- [EM Distillation for One-step Diffusion Models](https://mlsanity.com/paper/2405.16852/) (2024)
- [Your ViT is Secretly a Hybrid Discriminative-Generative Diffusion Model](https://mlsanity.com/paper/2208.07791/) (2022)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [On Sampling with Approximate Transport Maps](https://mlsanity.com/paper/2302.04763/) (2023)
- [End-to-End Stochastic Optimization with Energy-Based Model](https://mlsanity.com/paper/2211.13837/) (2022)
- [Guiding Energy-based Models via Contrastive Latent Variables](https://mlsanity.com/paper/2303.03023/) (2023)
- [Energy-Based Models for Anomaly Detection: A Manifold Diffusion Recovery Approach](https://mlsanity.com/paper/2310.18677/) (2023)
- [Explaining the effects of non-convergent sampling in the training of Energy-Based Models](https://mlsanity.com/paper/2301.09428/) (2023)
- [Revisiting Energy Based Models as Policies: Ranking Noise Contrastive Estimation and Interpolating Energy Models](https://mlsanity.com/paper/2309.05803/) (2023)
- [Hybrid Energy Based Model in the Feature Space for Out-of-Distribution Detection](https://mlsanity.com/paper/2305.16966/) (2023)
- [Fast training and sampling of Restricted Boltzmann Machines](https://mlsanity.com/paper/2405.15376/) (2024)
- [Fast and Functional Structured Data Generators Rooted in Out-of-Equilibrium Physics](https://mlsanity.com/paper/2307.06797/) (2023)
- [Noise-Free Sampling Algorithms via Regularized Wasserstein Proximals](https://mlsanity.com/paper/2308.14945/) (2023)
- [Energy based diffusion generator for efficient sampling of Boltzmann distributions](https://mlsanity.com/paper/2401.02080/) (2024)
- [Learning Latent Energy-Based Models via Interacting Particle Langevin Dynamics](https://mlsanity.com/paper/2510.12311/) (2025)
- [Balanced Training of Energy-Based Models with Adaptive Flow Sampling](https://mlsanity.com/paper/2306.00684/) (2023)
- [Training Latent Diffusion Models with Interacting Particle Algorithms](https://mlsanity.com/paper/2505.12412/) (2025)
- [Energy-Calibrated VAE with Test Time Free Lunch](https://mlsanity.com/paper/2311.04071/) (2023)
- [Latent Space Energy-based Model for Fine-grained Open Set Recognition](https://mlsanity.com/paper/2309.10711/) (2023)
- [Learning Energy-Based Generative Models via Potential Flow: A Variational Principle Approach to Probability Density Homotopy Matching](https://mlsanity.com/paper/2504.16262/) (2025)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [Hamiltonian Monte Carlo on ReLU Neural Networks is Inefficient](https://mlsanity.com/paper/2410.22065/) (2024)
- [Energy Correction Model in the Feature Space for Out-of-Distribution Detection](https://mlsanity.com/paper/2403.10403/) (2024)
- [Variational Potential Flow: A Novel Probabilistic Framework for Energy-Based Generative Modelling](https://mlsanity.com/paper/2407.15238/) (2024)
- [Recovering Hidden Reward in Diffusion-Based Policies](https://mlsanity.com/paper/2605.00623/) (2026)
- [ReFP-AD: Rectified Flow Preconditioning for Energy-Based Anomaly Detection](https://mlsanity.com/paper/2608.01793/) (2026)

## Abstract (arXiv)

Learning energy-based model (EBM) requires MCMC sampling of the learned model as an inner loop of the learning algorithm. However, MCMC sampling of EBMs in high-dimensional data space is generally not mixing, because the energy function, which is usually parametrized by a deep network, is highly multi-modal in the data space. This is a serious handicap for both theory and practice of EBMs. In this paper, we propose to learn an EBM with a flow-based model (or in general a latent variable model) serving as a backbone, so that the EBM is a correction or an exponential tilting of the flow-based model. We show that the model has a particularly simple form in the space of the latent variables of the backbone model, and MCMC sampling of the EBM in the latent space mixes well and traverses modes in the data space. This enables proper sampling and learning of EBMs.