# Principal Uncertainty Quantification with Spatial Correlation for Image Restoration Problems

arXiv: https://arxiv.org/abs/2305.10124 · HTML: https://mlsanity.com/paper/2305.10124/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 47 references that are in the map.
- First posted on arXiv 17 May 2023.

## Builds on (most cited first, 30 of 35)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Conformal Risk Control](https://mlsanity.com/paper/2208.02814/) (2022)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [Learn then Test: Calibrating Predictive Algorithms to Achieve Risk Control](https://mlsanity.com/paper/2110.01052/) (2021)
- [DRAW: A Recurrent Neural Network For Image Generation](https://mlsanity.com/paper/1502.04623/) (2015)
- [SNIPS: Solving Noisy Inverse Problems Stochastically](https://mlsanity.com/paper/2105.14951/) (2021)
- [Image-to-Image Regression with Distribution-Free Uncertainty Quantification and Applications in Imaging](https://mlsanity.com/paper/2202.05265/) (2022)
- [Distributional conformal prediction](https://mlsanity.com/paper/1909.07889/) (2019)
- [Attribute2Image: Conditional Image Generation from Visual Attributes](https://mlsanity.com/paper/1512.00570/) (2015)
- [Nested conformal prediction and quantile out-of-bag ensemble methods](https://mlsanity.com/paper/1910.10562/) (2019)
- [Solving Linear Inverse Problems Using the Prior Implicit in a Denoiser](https://mlsanity.com/paper/2007.13640/) (2020)
- [Conformal Prediction using Conditional Histograms](https://mlsanity.com/paper/2105.08747/) (2021)
- [Adaptive, Distribution-Free Prediction Intervals for Deep Networks](https://mlsanity.com/paper/1905.10634/) (2019)
- [How to Trust Your Diffusion Model: A Convex Optimization Approach to Conformal Risk Control](https://mlsanity.com/paper/2302.03791/) (2023)
- [A Comprehensive Analysis of Deep Regression](https://mlsanity.com/paper/1803.08450/) (2018)

## Built on by (most cited first, 28 of 28)

- [Fast yet Safe: Early-Exiting with Risk Control](https://mlsanity.com/paper/2405.20915/) (2024)
- [Conformal Contextual Robust Optimization](https://mlsanity.com/paper/2310.10003/) (2023)
- [Conformal Prediction: A Data Perspective](https://mlsanity.com/paper/2410.06494/) (2024)
- [Generative Uncertainty in Diffusion Models](https://mlsanity.com/paper/2502.20946/) (2025)
- [Uncertainty Quantification via Neural Posterior Principal Components](https://mlsanity.com/paper/2309.15533/) (2023)
- [Looks Too Good To Be True: An Information-Theoretic Analysis of Hallucinations in Generative Restoration Models](https://mlsanity.com/paper/2405.16475/) (2024)
- [InvFusion: Bridging Supervised and Zero-shot Diffusion for Inverse Problems](https://mlsanity.com/paper/2504.01689/) (2025)
- [Conformalized Link Prediction on Graph Neural Networks](https://mlsanity.com/paper/2406.18763/) (2024)
- [Plug-and-Play Posterior Sampling under Mismatched Measurement and Prior Models](https://mlsanity.com/paper/2310.03546/) (2023)
- [Uncertainty-Aware PPG-2-ECG for Enhanced Cardiovascular Diagnosis using Diffusion Models](https://mlsanity.com/paper/2405.11566/) (2024)
- [PSC: Posterior Sampling-Based Compression](https://mlsanity.com/paper/2407.09896/) (2024)
- [The Perception-Robustness Tradeoff in Deterministic Image Restoration](https://mlsanity.com/paper/2311.09253/) (2023)
- [Beyond Degradation Redundancy: Contrastive Prompt Learning for All-in-One Image Restoration](https://mlsanity.com/paper/2504.09973/) (2025)
- [WQLCP: Weighted Adaptive Conformal Prediction for Robust Uncertainty Quantification Under Distribution Shifts](https://mlsanity.com/paper/2505.19587/) (2025)
- [Hierarchical Uncertainty Exploration via Feedforward Posterior Trees](https://mlsanity.com/paper/2405.15719/) (2024)
- [COMPASS: Robust Feature Conformal Prediction for Medical Segmentation Metrics](https://mlsanity.com/paper/2509.22240/) (2025)
- [Uncertainty Visualization via Low-Dimensional Posterior Projections](https://mlsanity.com/paper/2312.07804/) (2023)
- [Metric-Guided Conformal Bounds for Probabilistic Image Reconstruction](https://mlsanity.com/paper/2404.15274/) (2024)
- [CONSIGN: Conformal Segmentation Informed by Spatial Groupings via Decomposition](https://mlsanity.com/paper/2505.14113/) (2025)
- [Conformal Bounds on Full-Reference Image Quality for Imaging Inverse Problems](https://mlsanity.com/paper/2505.09528/) (2025)
- [Segmentation-Guided CT Synthesis with Pixel-Wise Conformal Uncertainty Bounds](https://mlsanity.com/paper/2503.08515/) (2025)
- [Boosting All-in-One Image Restoration via Self-Improved Privilege Learning](https://mlsanity.com/paper/2505.24207/) (2025)
- [Decoding Federated Learning: The FedNAM+ Conformal Revolution](https://mlsanity.com/paper/2506.17872/) (2025)
- [Minimax Multi-Target Conformal Prediction with Applications to Imaging Inverse Problems](https://mlsanity.com/paper/2511.13533/) (2025)
- [CHEM: Estimating and Understanding Hallucinations in Deep Learning for Image Processing](https://mlsanity.com/paper/2512.09806/) (2025)
- [Assumption-free fidelity bounds for hardware noise characterization](https://mlsanity.com/paper/2504.07010/) (2025)
- [QUTCC: Quantile Uncertainty Training and Conformal Calibration for Imaging Inverse Problems](https://mlsanity.com/paper/2507.14760/) (2025)
- [Occupancy-based Quantile Risk Control](https://mlsanity.com/paper/2609.03104/) (2026)

## Abstract (arXiv)

Uncertainty quantification for inverse problems in imaging has drawn much attention lately. Existing approaches towards this task define uncertainty regions based on probable values per pixel, while ignoring spatial correlations within the image, resulting in an exaggerated volume of uncertainty. In this paper, we propose PUQ (Principal Uncertainty Quantification) -- a novel definition and corresponding analysis of uncertainty regions that takes into account spatial relationships within the image, thus providing reduced volume regions. Using recent advancements in generative models, we derive uncertainty intervals around principal components of the empirical posterior distribution, forming an ambiguity region that guarantees the inclusion of true unseen values with a user-defined confidence probability. To improve computational efficiency and interpretability, we also guarantee the recovery of true unseen values using only a few principal directions, resulting in more informative uncertainty regions. Our approach is verified through experiments on image colorization, super-resolution, and inpainting; its effectiveness is shown through comparison to baseline methods, demonstrating significantly tighter uncertainty regions.