# Reasons for the Superiority of Stochastic Estimators over Deterministic Ones: Robustness, Consistency and Perceptual Quality

arXiv: https://arxiv.org/abs/2211.08944 · HTML: https://mlsanity.com/paper/2211.08944/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 66 references that are in the map.
- First posted on arXiv 16 Nov 2022.

## Builds on (most cited first, 30 of 45)

- [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)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [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)
- [ESRGAN: Enhanced Super-Resolution Generative Adversarial Networks](https://mlsanity.com/paper/1809.00219/) (2018)
- [Image Super-Resolution Using Deep Convolutional Networks](https://mlsanity.com/paper/1501.00092/) (2014)
- [Real-Time Single Image and Video Super-Resolution Using an Efficient Sub-Pixel Convolutional Neural Network](https://mlsanity.com/paper/1609.05158/) (2016)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [Adversarial Machine Learning at Scale](https://mlsanity.com/paper/1611.01236/) (2016)
- [Accurate Image Super-Resolution Using Very Deep Convolutional Networks](https://mlsanity.com/paper/1511.04587/) (2015)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [Free-Form Image Inpainting with Gated Convolution](https://mlsanity.com/paper/1806.03589/) (2018)
- [The Perception-Distortion Tradeoff](https://mlsanity.com/paper/1711.06077/) (2017)
- [Generative Image Inpainting with Contextual Attention](https://mlsanity.com/paper/1801.07892/) (2018)
- [Assessing Generative Models via Precision and Recall](https://mlsanity.com/paper/1806.00035/) (2018)
- [A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets](https://mlsanity.com/paper/1707.08819/) (2017)
- [Deep multi-scale video prediction beyond mean square error](https://mlsanity.com/paper/1511.05440/) (2015)
- [Large Scale Image Completion via Co-Modulated Generative Adversarial Networks](https://mlsanity.com/paper/2103.10428/) (2021)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [SRFlow: Learning the Super-Resolution Space with Normalizing Flow](https://mlsanity.com/paper/2006.14200/) (2020)
- [SNIPS: Solving Noisy Inverse Problems Stochastically](https://mlsanity.com/paper/2105.14951/) (2021)
- [Contextual Residual Aggregation for Ultra High-Resolution Image Inpainting](https://mlsanity.com/paper/2005.09704/) (2020)

## Built on by (most cited first, 13 of 13)

- [Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --](https://mlsanity.com/paper/2301.03362/) (2023)
- [Idempotence and Perceptual Image Compression](https://mlsanity.com/paper/2401.08920/) (2024)
- [Compressed Image Generation with Denoising Diffusion Codebook Models](https://mlsanity.com/paper/2502.01189/) (2025)
- [Degradation-Aware Residual-Conditioned Optimal Transport for Unified Image Restoration](https://mlsanity.com/paper/2411.01656/) (2024)
- [Deep Optimal Transport: A Practical Algorithm for Photo-realistic Image Restoration](https://mlsanity.com/paper/2306.02342/) (2023)
- [High-Perceptual Quality JPEG Decoding via Posterior Sampling](https://mlsanity.com/paper/2211.11827/) (2022)
- [The Perception-Robustness Tradeoff in Deterministic Image Restoration](https://mlsanity.com/paper/2311.09253/) (2023)
- [Perceptual Fairness in Image Restoration](https://mlsanity.com/paper/2405.13805/) (2024)
- [pcaGAN: Improving Posterior-Sampling cGANs via Principal Component Regularization](https://mlsanity.com/paper/2411.00605/) (2024)
- [Double Blind Imaging with Generative Modeling](https://mlsanity.com/paper/2503.21501/) (2025)
- [Generative AI Meets 6G and Beyond: Diffusion Models for Semantic Communications](https://mlsanity.com/paper/2511.08416/) (2025)
- [Stochastic BIQA: Median Randomized Smoothing for Certified Blind Image Quality Assessment](https://mlsanity.com/paper/2411.12575/) (2024)
- [On Inverse Problems, Parameter Estimation, and Domain Generalization](https://mlsanity.com/paper/2506.06024/) (2025)

## Abstract (arXiv)

Stochastic restoration algorithms allow to explore the space of solutions that correspond to the degraded input. In this paper we reveal additional fundamental advantages of stochastic methods over deterministic ones, which further motivate their use. First, we prove that any restoration algorithm that attains perfect perceptual quality and whose outputs are consistent with the input must be a posterior sampler, and is thus required to be stochastic. Second, we illustrate that while deterministic restoration algorithms may attain high perceptual quality, this can be achieved only by filling up the space of all possible source images using an extremely sensitive mapping, which makes them highly vulnerable to adversarial attacks. Indeed, we show that enforcing deterministic models to be robust to such attacks profoundly hinders their perceptual quality, while robustifying stochastic models hardly influences their perceptual quality, and improves their output variability. These findings provide a motivation to foster progress in stochastic restoration methods, paving the way to better recovery algorithms.