mlsanityData to 14 Oct 2026

Path Integral Sampler: a stochastic control approach for sampling

arXiv 2111.15141 ↗First posted 30 Nov 2021cs.LGcited by 181

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Cited by 83 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

We present Path Integral Sampler~(PIS), a novel algorithm to draw samples from unnormalized probability density functions. The PIS is built on the Schr"odinger bridge problem which aims to recover the most likely evolution of a diffusion process given its initial distribution and terminal distribution. The PIS draws samples from the initial distribution and then propagates the samples through the Schr"odinger bridge to reach the terminal distribution. Applying the Girsanov theorem, with a simple prior diffusion, we formulate the PIS as a stochastic optimal control problem whose running cost is the control energy and terminal cost is chosen according to the target distribution. By modeling the control as a neural network, we establish a sampling algorithm that can be trained end-to-end. We provide theoretical justification of the sampling quality of PIS in terms of Wasserstein distance when sub-optimal control is used. Moreover, the path integrals theory is used to compute importance weights of the samples to compensate for the bias induced by the sub-optimality of the controller and time-discretization. We experimentally demonstrate the advantages of PIS compared with other start-of-the-art sampling methods on a variety of tasks.

Key facts. Cited by 181 papers in the map (data through 14 Oct 2026). Builds on 31 of its 63 references that are in the map. First posted on arXiv 30 Nov 2021.

Builds on

31 of 63 references.

References · 31

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  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  3. 2018Neural Ordinary Differential Equationscited by 3,779
  4. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,621
  5. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  6. 2015Variational Inference with Normalizing Flowscited by 2,170
  7. 2016Density estimation using Real NVPcited by 2,042
  8. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 594
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  10. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 292
  11. 2020Scalable Gradients for Stochastic Differential Equationscited by 271
  12. 2019Residual Flows for Invertible Generative Modelingcited by 257
19 more
  1. 2016A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluationcited by 236
  2. 2020Stochastic Normalizing Flowscited by 157
  3. 2019Learning Non-Convergent Non-Persistent Short-Run MCMC Toward Energy-Based Modelcited by 148
  4. 2016Learning in Implicit Generative Modelscited by 125
  5. 2020Learning Energy-Based Models by Diffusion Recovery Likelihoodcited by 123
  6. 2017Measuring Sample Quality with Kernelscited by 119
  7. 2014On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpointcited by 99
  8. 2019Theoretical guarantees for sampling and inference in generative models with latent diffusionscited by 97
  9. 2021Annealed Flow Transport Monte Carlocited by 90
  10. 2021Diffusion Normalizing Flowcited by 77
  11. 2019NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transportcited by 74
  12. 2021Efficient and Accurate Gradients for Neural SDEscited by 71
  13. 2017A-NICE-MC: Adversarial Training for MCMCcited by 38
  14. 2020i-flow: High-dimensional Integration and Sampling with Normalizing Flowscited by 26
  15. 2019Gradient-based Adaptive Markov Chain Monte Carlocited by 14
  16. 2014Path Integral Control and State Dependent Feedbackcited by 12
  17. 2020A Neural Network MCMC sampler that maximizes Proposal Entropycited by 12
  18. 2019Computational Separations between Sampling and Optimizationcited by 7
  19. 2020Deep Involutive Generative Models for Neural MCMCcited by 4

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 3; 2023: 15; 2024: 48; 2025: 64; 2026: 51.

181 papers.

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  1. 2023Improving and generalizing flow-based generative models with minibatch optimal transportcited by 879
  2. 2023I2SB: Image-to-Image Schr"odinger Bridgecited by 218
  3. 2024Semantic Image Inversion and Editing using Rectified Stochastic Differential Equationscited by 165
  4. 2023Denoising Diffusion Samplerscited by 149
  5. 2022An optimal control perspective on diffusion-based generative modelingcited by 147
  6. 2024Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Controlcited by 143
  7. 2024Iterated Denoising Energy Matching for Sampling from Boltzmann Densitiescited by 116
  8. 2023Improved sampling via learned diffusionscited by 104
  9. 2023A theory of continuous generative flow networkscited by 101
  10. 2024Understanding Reinforcement Learning-Based Fine-Tuning of Diffusion Models: A Tutorial and Reviewcited by 98
  11. 2023Unpaired Image-to-Image Translation via Neural Schr"odinger Bridgecited by 96
  12. 2025Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Expertscited by 96
38 more
  1. 2025Test-time Alignment of Diffusion Models without Reward Over-optimizationcited by 93
  2. 2024Physics-Informed Diffusion Modelscited by 90
  3. 2024Amortizing intractable inference in diffusion models for vision, language, and controlcited by 72
  4. 2024NETS: A Non-Equilibrium Transport Samplercited by 69
  5. 2024Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processescited by 66
  6. 2024Particle Denoising Diffusion Samplercited by 65
  7. 2025DIME:Diffusion-Based Maximum Entropy Reinforcement Learningcited by 58
  8. 2023Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimizationcited by 57
  9. 2024Beyond ELBOs: A Large-Scale Evaluation of Variational Methods for Samplingcited by 56
  10. 2024Symbolic Music Generation with Non-Differentiable Rule Guided Diffusioncited by 54
  11. 2024Target Score Matchingcited by 50
  12. 2024Improved off-policy training of diffusion samplerscited by 48
  13. 2024Sequential Controlled Langevin Diffusionscited by 48
  14. 2022Deep Generalized Schr"odinger Bridgecited by 47
  15. 2024DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transformcited by 46
  16. 2023Stochastic Optimal Control Matchingcited by 40
  17. 2023Aligned Diffusion Schr"odinger Bridgescited by 38
  18. 2025Scalable Equilibrium Sampling with Sequential Boltzmann Generatorscited by 38
  19. 2023SE(3) Equivariant Augmented Coupling Flowscited by 37
  20. 2025LEAPS: A discrete neural sampler via locally equivariant networkscited by 31
  21. 2023Blackout Diffusion: Generative Diffusion Models in Discrete-State Spacescited by 30
  22. 2025Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physicscited by 29
  23. 2025Underdamped Diffusion Bridges with Applications to Samplingcited by 29
  24. 2024Stochastic Localization via Iterative Posterior Samplingcited by 28
  25. 2025No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplerscited by 28
  26. 2025Progressive Inference-Time Annealing of Diffusion Models for Sampling from Boltzmann Densitiescited by 26
  27. 2024Liouville Flow Importance Samplercited by 25
  28. 2024Bridging Model-Based Optimization and Generative Modeling via Conservative Fine-Tuning of Diffusion Modelscited by 25
  29. 2024Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Samplingcited by 23
  30. 2024Adaptive teachers for amortized samplerscited by 23
  31. 2025Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inferencecited by 23
  32. 2025DriftLite: Lightweight Drift Control for Inference-Time Scaling of Diffusion Modelscited by 23
  33. 2023Transport meets Variational Inference: Controlled Monte Carlo Diffusionscited by 21
  34. 2024Training Neural Samplers with Reverse Diffusive KL Divergencecited by 21
  35. 2025Adjoint Schr"odinger Bridge Samplercited by 21
  36. 2025MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Controlcited by 21
  37. 2024Dynamical Measure Transport and Neural PDE Solvers for Samplingcited by 20
  38. 2024Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusioncited by 19

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