mlsanityData to 7 Oct 2026

How to Train Your Energy-Based Models

arXiv 2101.03288 ↗First posted 9 Jan 2021cs.LG stat.MLcited by 271

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Cited by 58 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

Energy-Based Models (EBMs), also known as non-normalized probabilistic models, specify probability density or mass functions up to an unknown normalizing constant. Unlike most other probabilistic models, EBMs do not place a restriction on the tractability of the normalizing constant, thus are more flexible to parameterize and can model a more expressive family of probability distributions. However, the unknown normalizing constant of EBMs makes training particularly difficult. Our goal is to provide a friendly introduction to modern approaches for EBM training. We start by explaining maximum likelihood training with Markov chain Monte Carlo (MCMC), and proceed to elaborate on MCMC-free approaches, including Score Matching (SM) and Noise Constrastive Estimation (NCE). We highlight theoretical connections among these three approaches, and end with a brief survey on alternative training methods, which are still under active research. Our tutorial is targeted at an audience with basic understanding of generative models who want to apply EBMs or start a research project in this direction.

Key facts. Cited by 271 papers in the map (data through 7 Oct 2026). Builds on 43 of its 85 references that are in the map. First posted on arXiv 9 Jan 2021.

Builds on

43 of 85 references.

References · 43

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  4. 2018Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actorcited by 4,635
  5. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  6. 2013Distributed Representations of Words and Phrases and their Compositionalitycited by 3,462
  7. 2015Variational Inference with Normalizing Flowscited by 2,161
  8. 2016Density estimation using Real NVPcited by 2,029
  9. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,339
  10. 2014NICE: Non-linear Independent Components Estimationcited by 1,052
  11. 2020Improved Techniques for Training Score-Based Generative Modelscited by 1,036
  12. 2016Pixel Recurrent Neural Networkscited by 814
31 more
  1. 2018MINE: Mutual Information Neural Estimationcited by 680
  2. 2017Reinforcement Learning with Deep Energy-Based Policiescited by 659
  3. 2016Improving Variational Inference with Inverse Autoregressive Flowcited by 636
  4. 2020WaveGrad: Estimating Gradients for Waveform Generationcited by 620
  5. 2013Generating Sequences With Recurrent Neural Networkscited by 549
  6. 2016f-GAN: Training Generative Neural Samplers using Variational Divergence Minimizationcited by 504
  7. 2018Spherical CNNscited by 452
  8. 2019Your Classifier is Secretly an Energy Based Model and You Should Treat it Like Onecited by 402
  9. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 381
  10. 2015MADE: Masked Autoencoder for Distribution Estimationcited by 306
  11. 2020Learning Gradient Fields for Shape Generationcited by 245
  12. 2016A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluationcited by 236
  13. 2016A Theory of Generative ConvNetcited by 174
  14. 2019Learning Non-Convergent Non-Persistent Short-Run MCMC Toward Energy-Based Modelcited by 147
  15. 2020Improved Contrastive Divergence Training of Energy Based Modelscited by 127
  16. 2020Residual Energy-Based Models for Text Generationcited by 95
  17. 2015Measuring Sample Quality with Stein's Methodcited by 94
  18. 2019Flow Contrastive Estimation of Energy-Based Modelscited by 75
  19. 2020No MCMC for me: Amortized sampling for fast and stable training of energy-based modelscited by 64
  20. 2019Maximum Entropy Generators for Energy-Based Modelscited by 55
  21. 2018Learning deep kernels for exponential family densitiescited by 36
  22. 2016Deep Directed Generative Models with Energy-Based Probability Estimationcited by 34
  23. 2019Exponential Family Estimation via Adversarial Dynamics Embeddingcited by 30
  24. 2020Training Deep Energy-Based Models with f-Divergence Minimizationcited by 29
  25. 2018Conditional Noise-Contrastive Estimation of Unnormalised Modelscited by 25
  26. 2018Adversarial Contrastive Estimationcited by 23
  27. 2018Kernel Exponential Family Estimation via Doubly Dual Embeddingcited by 16
  28. 2016Generative Adversarial Networks as Variational Training of Energy Based Modelscited by 10
  29. 2017Unbiased Markov chain Monte Carlo with couplingscited by 10
  30. 2020A Wasserstein Minimum Velocity Approach to Learning Unnormalized Modelscited by 9
  31. 2020Learning the Stein Discrepancy for Training and Evaluating Energy-Based Models without Samplingcited by 6

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 27; 2022: 38; 2023: 53; 2024: 63; 2025: 61; 2026: 29.

271 papers.

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  6. 2021Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecastingcited by 332
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  8. 2021Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Modelscited by 251
  9. 2021Pre-training Molecular Graph Representation with 3D Geometrycited by 248
  10. 2022A Survey on Generative Diffusion Modelcited by 238
  11. 2022Restoring Vision in Adverse Weather Conditions with Patch-Based Denoising Diffusion Modelscited by 215
  12. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matchingcited by 182
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  6. 2021GraphEBM: Molecular Graph Generation with Energy-Based Modelscited by 85
  7. 2023A Complete Survey on Generative AI (AIGC): Is ChatGPT from GPT-4 to GPT-5 All You Need?cited by 84
  8. 2023Two for One: Diffusion Models and Force Fields for Coarse-Grained Molecular Dynamicscited by 72
  9. 2023To Compress or Not to Compress- Self-Supervised Learning and Information Theory: A Reviewcited by 72
  10. 2024The Ingredients for Robotic Diffusion Transformerscited by 72
  11. 2022Molecular Geometry Pretraining with SE(3)-Invariant Denoising Distance Matchingcited by 67
  12. 2022MolGenSurvey: A Systematic Survey in Machine Learning Models for Molecule Designcited by 66
  13. 2021Controllable and Compositional Generation with Latent-Space Energy-Based Modelscited by 64
  14. 2021Estimating High Order Gradients of the Data Distribution by Denoisingcited by 63
  15. 2023A Text-guided Protein Design Frameworkcited by 61
  16. 2025Efficient Online Reinforcement Learning for Diffusion Policycited by 61
  17. 2024Optimal score estimation via empirical Bayes smoothingcited by 60
  18. 2023On the Design Fundamentals of Diffusion Models: A Surveycited by 59
  19. 2023CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problemscited by 54
  20. 2021Smooth Normalizing Flowscited by 52
  21. 2021Simulating Diffusion Bridges with Score Matchingcited by 51
  22. 2023Nonlinear Independent Component Analysis for Principled Disentanglement in Unsupervised Deep Learningcited by 48
  23. 2022Probability flow solution of the Fokker-Planck equationcited by 46
  24. 2021ScoreGrad: Multivariate Probabilistic Time Series Forecasting with Continuous Energy-based Generative Modelscited by 42
  25. 2025Is Noise Conditioning Necessary for Denoising Generative Models?cited by 42
  26. 2023Diffusion Model-Augmented Behavioral Cloningcited by 37
  27. 2022Flow-matching -- efficient coarse-graining of molecular dynamics without forcescited by 36
  28. 2024Deep Generative Models in Robotics: A Survey on Learning from Multimodal Demonstrationscited by 35
  29. 2024Denoising: A Powerful Building-Block for Imaging, Inverse Problems, and Machine Learningcited by 35
  30. 2023A Group Symmetric Stochastic Differential Equation Model for Molecule Multi-modal Pretrainingcited by 32
  31. 2023Shelving, Stacking, Hanging: Relational Pose Diffusion for Multi-modal Rearrangementcited by 29
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  33. 2022Learning to Solve Travelling Salesman Problem with Hardness-adaptive Curriculumcited by 27
  34. 2021Autoencoding Under Normalization Constraintscited by 26
  35. 2023Energy-guided Entropic Neural Optimal Transportcited by 25
  36. 2023TEA: Test-time Energy Adaptationcited by 25
  37. 2024Training Unbiased Diffusion Models From Biased Datasetcited by 25
  38. 2025Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modelingcited by 25

50 most cited of 271 All 271 (JSON)