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Stochastic Normalizing Flows for Inverse Problems: a Markov Chains Viewpoint

arXiv 2109.11375 ↗First posted 23 Sep 2021cs.LG math.PRcited by 32

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Abstract (arXiv)

To overcome topological constraints and improve the expressiveness of normalizing flow architectures, Wu, K"ohler and No'e introduced stochastic normalizing flows which combine deterministic, learnable flow transformations with stochastic sampling methods. In this paper, we consider stochastic normalizing flows from a Markov chain point of view. In particular, we replace transition densities by general Markov kernels and establish proofs via Radon-Nikodym derivatives which allows to incorporate distributions without densities in a sound way. Further, we generalize the results for sampling from posterior distributions as required in inverse problems. The performance of the proposed conditional stochastic normalizing flow is demonstrated by numerical examples.

Key facts. Cited by 32 papers in the map (data through 7 Oct 2026). Builds on 32 of its 55 references that are in the map. First posted on arXiv 23 Sep 2021.

Builds on

32 of 55 references.

References · 32

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  4. 2015Variational Inference with Normalizing Flowscited by 2,161
  5. 2016Density estimation using Real NVPcited by 2,029
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  10. 2020Deep Learning Techniques for Inverse Problems in Imagingcited by 290
  11. 2019Residual Flows for Invertible Generative Modelingcited by 256
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20 more
  1. 2018Analyzing Inverse Problems with Invertible Neural Networkscited by 239
  2. 2018Neural Autoregressive Flowscited by 207
  3. 2020Stochastic Normalizing Flowscited by 156
  4. 2018Neural Importance Samplingcited by 154
  5. 2019Guided Image Generation with Conditional Invertible Neural Networkscited by 147
  6. 2021An Introduction to Deep Generative Modelingcited by 104
  7. 2021Annealed Flow Transport Monte Carlocited by 90
  8. 2020Understanding and Mitigating Exploding Inverses in Invertible Neural Networkscited by 71
  9. 2020SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flowscited by 71
  10. 2018Explorations in Homeomorphic Variational Auto-Encodingcited by 65
  11. 2019Block Neural Autoregressive Flowcited by 62
  12. 2020Composing Normalizing Flows for Inverse Problemscited by 48
  13. 2017A-NICE-MC: Adversarial Training for MCMCcited by 36
  14. 2021Monte Carlo Variational Auto-Encoderscited by 36
  15. 2019HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inferencecited by 31
  16. 2020Stabilizing Invertible Neural Networks Using Mixture Modelscited by 30
  17. 2021Conditional Invertible Neural Networks for Diverse Image-to-Image Translationcited by 24
  18. 2017Continuous-Time Flows for Efficient Inference and Density Estimationcited by 21
  19. 2021Invertible Neural Networks versus MCMC for Posterior Reconstruction in Grazing Incidence X-Ray Fluorescencecited by 12
  20. 2021Multiscale Invertible Generative Networks for High-Dimensional Bayesian Inferencecited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 4; 2023: 11; 2024: 6; 2025: 7; 2026: 3.

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  1. 2023A theory of continuous generative flow networkscited by 98
  2. 2023Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generationcited by 37
  3. 2024Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matchingcited by 34
  4. 2023Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernelcited by 25
  5. 2021Generalized Normalizing Flows via Markov Chainscited by 23
  6. 2025Flower: A Flow-Matching Solver for Inverse Problemscited by 23
  7. 2022PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularizationcited by 22
  8. 2025Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approachcited by 18
  9. 2023Manifold Learning by Mixture Models of VAEs for Inverse Problemscited by 16
  10. 2025On the Relation between Rectified Flows and Optimal Transportcited by 15
  11. 2023Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problemscited by 12
  12. 2023A Review of Change of Variable Formulas for Generative Modelingcited by 12
20 more
  1. 2024Annealing Flow Generative Models Towards Sampling High-Dimensional and Multi-Modal Distributionscited by 11
  2. 2022WPPNets and WPPFlows: The Power of Wasserstein Patch Priors for Superresolutioncited by 10
  3. 2023Neural Wasserstein Gradient Flows for Maximum Mean Discrepancies with Riesz Kernelscited by 9
  4. 2023Learning from small data sets: Patch-based regularizers in inverse problems for image reconstructioncited by 9
  5. 2024Importance Corrected Neural JKO Samplingcited by 7
  6. 2022Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spacescited by 6
  7. 2022Proximal Residual Flows for Bayesian Inverse Problemscited by 5
  8. 2024Stability of Data-Dependent Ridge-Regularization for Inverse Problemscited by 5
  9. 2023NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problemscited by 4
  10. 2023Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distancescited by 3
  11. 2025Diffusion Models with Double Guidance: Generate with aggregated datasetscited by 2
  12. 2025A Mathematical Explanation of Transformerscited by 2
  13. 2026Conditional flow matching for physics-constrained inverse problems with finite training datacited by 2
  14. 2024Mixed Noise and Posterior Estimation with Conditional DeepGEMcited by 1
  15. 2025A Constraint-Preserving Neural Network Approach for Solving Mean-Field Games Equilibriumcited by 1
  16. 2025An invertible generative model for forward and inverse problemscited by 1
  17. 2023Learning variational autoencoders via MCMC speed measurescited by 0
  18. 2024Sampling from Boltzmann densities with physics informed low-rank formatscited by 0
  19. 2026Sampling via Stochastic Interpolants by Langevin-based Velocity and Initialization Estimation in Flow ODEscited by 0
  20. 2026Deep unfolding of MCMC kernels: scalable, modular & explainable GANs for high-dimensional posterior samplingcited by 0