# Composing Normalizing Flows for Inverse Problems

arXiv: https://arxiv.org/abs/2002.11743 · HTML: https://mlsanity.com/paper/2002.11743/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2020.

## Built on by (most cited first, 30 of 48)

- [Deblurring via Stochastic Refinement](https://mlsanity.com/paper/2112.02475/) (2021)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Training-free Linear Image Inverses via Flows](https://mlsanity.com/paper/2310.04432/) (2023)
- [Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching](https://mlsanity.com/paper/2405.18816/) (2024)
- [WaveDM: Wavelet-Based Diffusion Models for Image Restoration](https://mlsanity.com/paper/2305.13819/) (2023)
- [On Exact Inversion of DPM-Solvers](https://mlsanity.com/paper/2311.18387/) (2023)
- [Stochastic Normalizing Flows for Inverse Problems: a Markov Chains Viewpoint](https://mlsanity.com/paper/2109.11375/) (2021)
- [Generative Visual Prompt: Unifying Distributional Control of Pre-Trained Generative Models](https://mlsanity.com/paper/2209.06970/) (2022)
- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [The efficacy and generalizability of conditional GANs for posterior inference in physics-based inverse problems](https://mlsanity.com/paper/2202.07773/) (2022)
- [Provable Compressed Sensing with Generative Priors via Langevin Dynamics](https://mlsanity.com/paper/2102.12643/) (2021)
- [Conditional Injective Flows for Bayesian Imaging](https://mlsanity.com/paper/2204.07664/) (2022)
- [Gradual Domain Adaptation via Normalizing Flows](https://mlsanity.com/paper/2206.11492/) (2022)
- [ASPIRE: Iterative Amortized Posterior Inference for Bayesian Inverse Problems](https://mlsanity.com/paper/2405.05398/) (2024)
- [VQ-Flows: Vector Quantized Local Normalizing Flows](https://mlsanity.com/paper/2203.11556/) (2022)
- [FrePolad: Frequency-Rectified Point Latent Diffusion for Point Cloud Generation](https://mlsanity.com/paper/2311.12090/) (2023)
- [Universal Functional Regression with Neural Operator Flows](https://mlsanity.com/paper/2404.02986/) (2024)
- [Amortized Normalizing Flows for Transcranial Ultrasound with Uncertainty Quantification](https://mlsanity.com/paper/2303.03478/) (2023)
- [A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing](https://mlsanity.com/paper/2310.03758/) (2023)
- [Optimization for Amortized Inverse Problems](https://mlsanity.com/paper/2210.13983/) (2022)
- [Conditional score-based diffusion models for solving inverse problems in mechanics](https://mlsanity.com/paper/2406.13154/) (2024)
- [Conformalized Generative Bayesian Imaging: An Uncertainty Quantification Framework for Computational Imaging](https://mlsanity.com/paper/2504.07696/) (2025)
- [Posterior Matching for Arbitrary Conditioning](https://mlsanity.com/paper/2201.12414/) (2022)
- [Iterative Flow Matching: Path Correction and Gradual Refinement for Enhanced Generative Modeling](https://mlsanity.com/paper/2502.16445/) (2025)
- [AdaAnn: Adaptive Annealing Scheduler for Probability Density Approximation](https://mlsanity.com/paper/2202.00792/) (2022)
- [Equivariant Priors for Compressed Sensing with Unknown Orientation](https://mlsanity.com/paper/2206.14069/) (2022)
- [Deep Injective Prior for Inverse Scattering](https://mlsanity.com/paper/2301.03092/) (2023)
- [Accelerating Inverse Learning via Intelligent Localization with Exploratory Sampling](https://mlsanity.com/paper/2212.01016/) (2022)
- [Deep Variational Inverse Scattering](https://mlsanity.com/paper/2212.04309/) (2022)

## Abstract (arXiv)

Given an inverse problem with a normalizing flow prior, we wish to estimate the distribution of the underlying signal conditioned on the observations. We approach this problem as a task of conditional inference on the pre-trained unconditional flow model. We first establish that this is computationally hard for a large class of flow models. Motivated by this, we propose a framework for approximate inference that estimates the target conditional as a composition of two flow models. This formulation leads to a stable variational inference training procedure that avoids adversarial training. Our method is evaluated on a variety of inverse problems and is shown to produce high-quality samples with uncertainty quantification. We further demonstrate that our approach can be amortized for zero-shot inference.