# Stochastic Normalizing Flows for Inverse Problems: a Markov Chains Viewpoint

arXiv: https://arxiv.org/abs/2109.11375 · HTML: https://mlsanity.com/paper/2109.11375/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 55 references that are in the map.
- First posted on arXiv 23 Sep 2021.

## Builds on (most cited first, 30 of 32)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [cGANs with Projection Discriminator](https://mlsanity.com/paper/1802.05637/) (2018)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Guided Image Generation with Conditional Invertible Neural Networks](https://mlsanity.com/paper/1907.02392/) (2019)
- [An Introduction to Deep Generative Modeling](https://mlsanity.com/paper/2103.05180/) (2021)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Understanding and Mitigating Exploding Inverses in Invertible Neural Networks](https://mlsanity.com/paper/2006.09347/) (2020)
- [SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows](https://mlsanity.com/paper/2007.02731/) (2020)
- [Explorations in Homeomorphic Variational Auto-Encoding](https://mlsanity.com/paper/1807.04689/) (2018)
- [Block Neural Autoregressive Flow](https://mlsanity.com/paper/1904.04676/) (2019)
- [Composing Normalizing Flows for Inverse Problems](https://mlsanity.com/paper/2002.11743/) (2020)
- [A-NICE-MC: Adversarial Training for MCMC](https://mlsanity.com/paper/1706.07561/) (2017)
- [Monte Carlo Variational Auto-Encoders](https://mlsanity.com/paper/2106.15921/) (2021)
- [HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference](https://mlsanity.com/paper/1905.10687/) (2019)
- [Stabilizing Invertible Neural Networks Using Mixture Models](https://mlsanity.com/paper/2009.02994/) (2020)
- [Conditional Invertible Neural Networks for Diverse Image-to-Image Translation](https://mlsanity.com/paper/2105.02104/) (2021)
- [Continuous-Time Flows for Efficient Inference and Density Estimation](https://mlsanity.com/paper/1709.01179/) (2017)

## Built on by (most cited first, 30 of 32)

- [A theory of continuous generative flow networks](https://mlsanity.com/paper/2301.12594/) (2023)
- [Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation](https://mlsanity.com/paper/2303.04772/) (2023)
- [Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching](https://mlsanity.com/paper/2403.18705/) (2024)
- [Flower: A Flow-Matching Solver for Inverse Problems](https://mlsanity.com/paper/2509.26287/) (2025)
- [Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel](https://mlsanity.com/paper/2310.03054/) (2023)
- [Generalized Normalizing Flows via Markov Chains](https://mlsanity.com/paper/2111.12506/) (2021)
- [PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularization](https://mlsanity.com/paper/2205.12021/) (2022)
- [On the Relation between Rectified Flows and Optimal Transport](https://mlsanity.com/paper/2505.19712/) (2025)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Manifold Learning by Mixture Models of VAEs for Inverse Problems](https://mlsanity.com/paper/2303.15244/) (2023)
- [Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problems](https://mlsanity.com/paper/2303.15845/) (2023)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Annealing Flow Generative Models Towards Sampling High-Dimensional and Multi-Modal Distributions](https://mlsanity.com/paper/2409.20547/) (2024)
- [WPPNets and WPPFlows: The Power of Wasserstein Patch Priors for Superresolution](https://mlsanity.com/paper/2201.08157/) (2022)
- [Neural Wasserstein Gradient Flows for Maximum Mean Discrepancies with Riesz Kernels](https://mlsanity.com/paper/2301.11624/) (2023)
- [Learning from small data sets: Patch-based regularizers in inverse problems for image reconstruction](https://mlsanity.com/paper/2312.16611/) (2023)
- [Importance Corrected Neural JKO Sampling](https://mlsanity.com/paper/2407.20444/) (2024)
- [Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces](https://mlsanity.com/paper/2205.14627/) (2022)
- [Proximal Residual Flows for Bayesian Inverse Problems](https://mlsanity.com/paper/2211.17158/) (2022)
- [Stability of Data-Dependent Ridge-Regularization for Inverse Problems](https://mlsanity.com/paper/2406.12289/) (2024)
- [NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems](https://mlsanity.com/paper/2304.08342/) (2023)
- [Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distances](https://mlsanity.com/paper/2310.13433/) (2023)
- [Diffusion Models with Double Guidance: Generate with aggregated datasets](https://mlsanity.com/paper/2505.13213/) (2025)
- [An invertible generative model for forward and inverse problems](https://mlsanity.com/paper/2509.03910/) (2025)
- [A Mathematical Explanation of Transformers](https://mlsanity.com/paper/2510.03989/) (2025)
- [Conditional flow matching for physics-constrained inverse problems with finite training data](https://mlsanity.com/paper/2603.14135/) (2026)
- [Mixed Noise and Posterior Estimation with Conditional DeepGEM](https://mlsanity.com/paper/2402.02964/) (2024)
- [A Constraint-Preserving Neural Network Approach for Solving Mean-Field Games Equilibrium](https://mlsanity.com/paper/2501.17450/) (2025)
- [Learning variational autoencoders via MCMC speed measures](https://mlsanity.com/paper/2308.13731/) (2023)
- [Sampling from Boltzmann densities with physics informed low-rank formats](https://mlsanity.com/paper/2412.07637/) (2024)

## 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.