mlsanityData to 7 Oct 2026

Transport meets Variational Inference: Controlled Monte Carlo Diffusions

arXiv 2307.01050 ↗First posted 3 Jul 2023stat.ML cs.LGcited by 21

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Abstract (arXiv)

Connecting optimal transport and variational inference, we present a principled and systematic framework for sampling and generative modelling centred around divergences on path space. Our work culminates in the development of the Controlled Monte Carlo Diffusion sampler (CMCD) for Bayesian computation, a score-based annealing technique that crucially adapts both forward and backward dynamics in a diffusion model. On the way, we clarify the relationship between the EM-algorithm and iterative proportional fitting (IPF) for Schr"odinger bridges, deriving as well a regularised objective that bypasses the iterative bottleneck of standard IPF-updates. Finally, we show that CMCD has a strong foundation in the Jarzinsky and Crooks identities from statistical physics, and that it convincingly outperforms competing approaches across a wide array of experiments.

Key facts. Cited by 21 papers in the map (data through 7 Oct 2026). Builds on 55 of its 132 references that are in the map. First posted on arXiv 3 Jul 2023.

Builds on

55 of 132 references.

References · 55

  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. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  5. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,277
  6. 2015Variational Inference with Normalizing Flowscited by 2,161
  7. 2016Variational Inference: A Review for Statisticianscited by 1,466
  8. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  9. 2021Variational Diffusion Modelscited by 1,145
  10. 2019An Introduction to Variational Autoencoderscited by 997
  11. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  12. 2017Masked Autoregressive Flow for Density Estimationcited by 680
38 more
  1. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modelingcited by 594
  2. 2015Importance Weighted Autoencoderscited by 454
  3. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 381
  4. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 286
  5. 2021GFlowNet Foundationscited by 286
  6. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 216
  7. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matchingcited by 182
  8. 2021Path Integral Sampler: a stochastic control approach for samplingcited by 175
  9. 2019Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limitcited by 168
  10. 2020Stochastic Normalizing Flowscited by 156
  11. 2022An optimal control perspective on diffusion-based generative modelingcited by 145
  12. 2023Denoising Diffusion Samplerscited by 144
  13. 2020OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transportcited by 143
  14. 2021Solving Schr"odinger Bridges via Maximum Likelihoodcited by 140
  15. 2022Flow Annealed Importance Sampling Bootstrapcited by 124
  16. 2016R'enyi Divergence Variational Inferencecited by 104
  17. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Modelscited by 102
  18. 2014On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpointcited by 99
  19. 2017Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inferencecited by 97
  20. 2019Theoretical guarantees for sampling and inference in generative models with latent diffusionscited by 97
  21. 2022Action Matching: Learning Stochastic Dynamics from Samplescited by 94
  22. 2021Annealed Flow Transport Monte Carlocited by 90
  23. 2020Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path spacecited by 80
  24. 2023Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modelingcited by 80
  25. 2021Diffusion Normalizing Flowcited by 77
  26. 2020VarGrad: A Low-Variance Gradient Estimator for Variational Inferencecited by 73
  27. 2022Continual Repeated Annealed Flow Transport Monte Carlocited by 66
  28. 2021Bayesian Learning via Neural Schr"odinger-F"ollmer Flowscited by 56
  29. 2022Score-Based Diffusion meets Annealed Importance Samplingcited by 51
  30. 2022Entropic Neural Optimal Transport via Diffusion Processescited by 49
  31. 2021MCMC Variational Inference via Uncorrected Hamiltonian Annealingcited by 39
  32. 2021Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equationscited by 38
  33. 2015Black-box α-divergence Minimizationcited by 37
  34. 2021Monte Carlo Variational Auto-Encoderscited by 36
  35. 2021Differentiable Annealed Importance Sampling and the Perils of Gradient Noisecited by 36
  36. 2023A mean-field games laboratory for generative modelingcited by 28
  37. 2022Langevin Diffusion Variational Inferencecited by 26
  38. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21

50 most cited of 55 All 55 (JSON)

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 3; 2025: 13; 2026: 5.

21 papers.

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  1. 2025Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Expertscited by 85
  2. 2025DIME:Diffusion-Based Maximum Entropy Reinforcement Learningcited by 52
  3. 2025Any-Order Flexible Length Masked Diffusioncited by 48
  4. 2025Scalable Equilibrium Sampling with Sequential Boltzmann Generatorscited by 36
  5. 2025LEAPS: A discrete neural sampler via locally equivariant networkscited by 30
  6. 2025Debiasing Guidance for Discrete Diffusion with Sequential Monte Carlocited by 29
  7. 2024Stochastic Localization via Iterative Posterior Samplingcited by 27
  8. 2025DriftLite: Lightweight Drift Control for Inference-Time Scaling of Diffusion Modelscited by 21
  9. 2025Tilt Matching for Scalable Sampling and Fine-Tuningcited by 17
  10. 2025Outsourced diffusion sampling: Efficient posterior inference in latent spaces of generative modelscited by 16
  11. 2025Temperature-Annealed Boltzmann Generatorscited by 14
  12. 2025Progressive Tempering Sampler with Diffusioncited by 13
9 more
  1. 2024Importance Corrected Neural JKO Samplingcited by 7
  2. 2026Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matchingcited by 7
  3. 2025Control Consistency Losses for Diffusion Bridgescited by 6
  4. 2025Reinforced sequential Monte Carlo for amortised samplingcited by 5
  5. 2026Flow Sampling: Learning to Sample from Unnormalized Densities via Denoising Conditional Processescited by 4
  6. 2026Discrete diffusion samplers and bridges: Off-policy algorithms and applications in latent spacescited by 2
  7. 2026Counterdiabatic Hamiltonian Monte Carlocited by 1
  8. 2026Conditional Diffusion Samplingcited by 1
  9. 2024Mirror Bridges Between Probability Measurescited by 0

Similar, not cited

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  1. 2026Diffusion Path Samplers via Sequential Monte Carlo
  2. 2023FreeFlow: A Comprehensive Understanding on Diffusion Probabilistic Models via Optimal Transport
  3. 2026Simple Approximation and Derivative Free Inference-Time Scaling for Diffusion Models via Sequential Monte Carlo on Path Measures
  4. 2025Briding Diffusion Posterior Sampling and Monte Carlo methods: a survey
  5. 2025Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physics
  6. 2023Accelerating Markov Chain Monte Carlo sampling with diffusion models
  7. 2025Variational Control for Guidance in Diffusion Models
  8. 2026Composing diffusion priors with explicit physical context via generative Gibbs sampling