# On the Robustness of Normalizing Flows for Inverse Problems in Imaging

arXiv: https://arxiv.org/abs/2212.04319 · HTML: https://mlsanity.com/paper/2212.04319/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 54 references that are in the map.
- First posted on arXiv 8 Dec 2022.

## Builds on (most cited first, 30 of 38)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [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)
- [A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks](https://mlsanity.com/paper/1807.03888/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [ESRGAN: Enhanced Super-Resolution Generative Adversarial Networks](https://mlsanity.com/paper/1809.00219/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Deep Retinex Decomposition for Low-Light Enhancement](https://mlsanity.com/paper/1808.04560/) (2018)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [WaveGlow: A Flow-based Generative Network for Speech Synthesis](https://mlsanity.com/paper/1811.00002/) (2018)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Why Normalizing Flows Fail to Detect Out-of-Distribution Data](https://mlsanity.com/paper/2006.08545/) (2020)
- [SRFlow: Learning the Super-Resolution Space with Normalizing Flow](https://mlsanity.com/paper/2006.14200/) (2020)
- [Low-Light Image Enhancement with Normalizing Flow](https://mlsanity.com/paper/2109.05923/) (2021)
- [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)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Noise Flow: Noise Modeling with Conditional Normalizing Flows](https://mlsanity.com/paper/1908.08453/) (2019)
- [Hierarchical Conditional Flow: A Unified Framework for Image Super-Resolution and Image Rescaling](https://mlsanity.com/paper/2108.05301/) (2021)
- [Understanding and Mitigating Exploding Inverses in Invertible Neural Networks](https://mlsanity.com/paper/2006.09347/) (2020)
- [Solving Inverse Problems With Deep Neural Networks -- Robustness Included?](https://mlsanity.com/paper/2011.04268/) (2020)
- [Smooth Normalizing Flows](https://mlsanity.com/paper/2110.00351/) (2021)
- [C-Flow: Conditional Generative Flow Models for Images and 3D Point Clouds](https://mlsanity.com/paper/1912.07009/) (2019)

## Built on by (most cited first, 8 of 8)

- [Training-free Linear Image Inverses via Flows](https://mlsanity.com/paper/2310.04432/) (2023)
- [GLARE: Low Light Image Enhancement via Generative Latent Feature based Codebook Retrieval](https://mlsanity.com/paper/2407.12431/) (2024)
- [Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problems](https://mlsanity.com/paper/2303.15845/) (2023)
- [Boosting Flow-based Generative Super-Resolution Models via Learned Prior](https://mlsanity.com/paper/2403.10988/) (2024)
- [Adaptive Selection of Sampling-Reconstruction in Fourier Compressed Sensing](https://mlsanity.com/paper/2409.11738/) (2024)
- [Constructing Fair Latent Space for Intersection of Fairness and Explainability](https://mlsanity.com/paper/2412.17523/) (2024)
- [Analytic Bijections for Smooth and Interpretable Normalizing Flows](https://mlsanity.com/paper/2601.10774/) (2026)
- [ShuffleFlow: Scalable Posterior Inference for Bayesian Inverse Imaging](https://mlsanity.com/paper/2606.21099/) (2026)

## Abstract (arXiv)

Conditional normalizing flows can generate diverse image samples for solving inverse problems. Most normalizing flows for inverse problems in imaging employ the conditional affine coupling layer that can generate diverse images quickly. However, unintended severe artifacts are occasionally observed in the output of them. In this work, we address this critical issue by investigating the origins of these artifacts and proposing the conditions to avoid them. First of all, we empirically and theoretically reveal that these problems are caused by "exploding inverse" in the conditional affine coupling layer for certain out-of-distribution (OOD) conditional inputs. Then, we further validated that the probability of causing erroneous artifacts in pixels is highly correlated with a Mahalanobis distance-based OOD score for inverse problems in imaging. Lastly, based on our investigations, we propose a remark to avoid exploding inverse and then based on it, we suggest a simple remedy that substitutes the affine coupling layers with the modified rational quadratic spline coupling layers in normalizing flows, to encourage the robustness of generated image samples. Our experimental results demonstrated that our suggested methods effectively suppressed critical artifacts occurring in normalizing flows for super-resolution space generation and low-light image enhancement.