# A Theory of the Distortion-Perception Tradeoff in Wasserstein Space

arXiv: https://arxiv.org/abs/2107.02555 · HTML: https://mlsanity.com/paper/2107.02555/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 36 references that are in the map.
- First posted on arXiv 6 Jul 2021.

## Builds on (most cited first, 21 of 21)

- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [Enhanced Deep Residual Networks for Single Image Super-Resolution](https://mlsanity.com/paper/1707.02921/) (2017)
- [ESRGAN: Enhanced Super-Resolution Generative Adversarial Networks](https://mlsanity.com/paper/1809.00219/) (2018)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [The Perception-Distortion Tradeoff](https://mlsanity.com/paper/1711.06077/) (2017)
- [SinGAN: Learning a Generative Model from a Single Natural Image](https://mlsanity.com/paper/1905.01164/) (2019)
- ["Zero-Shot" Super-Resolution using Deep Internal Learning](https://mlsanity.com/paper/1712.06087/) (2017)
- [Rethinking Lossy Compression: The Rate-Distortion-Perception Tradeoff](https://mlsanity.com/paper/1901.07821/) (2019)
- [SRFlow: Learning the Super-Resolution Space with Normalizing Flow](https://mlsanity.com/paper/2006.14200/) (2020)
- [EnhanceNet: Single Image Super-Resolution Through Automated Texture Synthesis](https://mlsanity.com/paper/1612.07919/) (2016)
- [The 2018 PIRM Challenge on Perceptual Image Super-resolution](https://mlsanity.com/paper/1809.07517/) (2018)
- [Deep Network Interpolation for Continuous Imagery Effect Transition](https://mlsanity.com/paper/1811.10515/) (2018)
- [Stochastic Image Denoising by Sampling from the Posterior Distribution](https://mlsanity.com/paper/2101.09552/) (2021)
- [Explorable Super Resolution](https://mlsanity.com/paper/1912.01839/) (2019)
- [High Perceptual Quality Image Denoising with a Posterior Sampling CGAN](https://mlsanity.com/paper/2103.04192/) (2021)
- [On The Classification-Distortion-Perception Tradeoff](https://mlsanity.com/paper/1904.08816/) (2019)
- [Dynamic-Net: Tuning the Objective Without Re-training for Synthesis Tasks](https://mlsanity.com/paper/1811.08760/) (2018)
- [Fully Unsupervised Diversity Denoising with Convolutional Variational Autoencoders](https://mlsanity.com/paper/2006.06072/) (2020)
- [Posterior Sampling for Image Restoration using Explicit Patch Priors](https://mlsanity.com/paper/2104.09895/) (2021)
- [Multi-Scale Recursive and Perception-Distortion Controllable Image Super-Resolution](https://mlsanity.com/paper/1809.10711/) (2018)
- [A Generative Model for Hallucinating Diverse Versions of Super Resolution Images](https://mlsanity.com/paper/2102.06624/) (2021)

## Built on by (most cited first, 30 of 52)

- [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)
- [Posterior-Mean Rectified Flow: Towards Minimum MSE Photo-Realistic Image Restoration](https://mlsanity.com/paper/2410.00418/) (2024)
- [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)
- [Perception-Distortion Balanced ADMM Optimization for Single-Image Super-Resolution](https://mlsanity.com/paper/2208.03324/) (2022)
- [SILO: Solving Inverse Problems with Latent Operators](https://mlsanity.com/paper/2501.11746/) (2025)
- [ACDMSR: Accelerated Conditional Diffusion Models for Single Image Super-Resolution](https://mlsanity.com/paper/2307.00781/) (2023)
- [Deep Optimal Transport: A Practical Algorithm for Photo-realistic Image Restoration](https://mlsanity.com/paper/2306.02342/) (2023)
- [Optimally Controllable Perceptual Lossy Compression](https://mlsanity.com/paper/2206.10082/) (2022)
- [Reasons for the Superiority of Stochastic Estimators over Deterministic Ones: Robustness, Consistency and Perceptual Quality](https://mlsanity.com/paper/2211.08944/) (2022)
- [Looks Too Good To Be True: An Information-Theoretic Analysis of Hallucinations in Generative Restoration Models](https://mlsanity.com/paper/2405.16475/) (2024)
- [On the Choice of Perception Loss Function for Learned Video Compression](https://mlsanity.com/paper/2305.19301/) (2023)
- [Rate-Distortion-Perception Tradeoff Based on the Conditional-Distribution Perception Measure](https://mlsanity.com/paper/2401.12207/) (2024)
- [Efficient Image Restoration via Latent Consistency Flow Matching](https://mlsanity.com/paper/2502.03500/) (2025)
- [The Perception-Robustness Tradeoff in Deterministic Image Restoration](https://mlsanity.com/paper/2311.09253/) (2023)
- [Uncovering the Over-smoothing Challenge in Image Super-Resolution: Entropy-based Quantification and Contrastive Optimization](https://mlsanity.com/paper/2201.01034/) (2022)
- [Conditional Perceptual Quality Preserving Image Compression](https://mlsanity.com/paper/2308.08154/) (2023)
- [Reconciling Stochastic and Deterministic Strategies for Zero-shot Image Restoration using Diffusion Model in Dual](https://mlsanity.com/paper/2503.01288/) (2025)
- [Traversing Distortion-Perception Tradeoff using a Single Score-Based Generative Model](https://mlsanity.com/paper/2503.20297/) (2025)
- [Gaussian Rate-Distortion-Perception Coding and Entropy-Constrained Scalar Quantization](https://mlsanity.com/paper/2409.02388/) (2024)
- [Prior-Informed Flow Matching for Graph Reconstruction](https://mlsanity.com/paper/2601.22107/) (2026)
- [Perceptual Kalman Filters: Online State Estimation under a Perfect Perceptual-Quality Constraint](https://mlsanity.com/paper/2306.02400/) (2023)
- [A New Perspective On Denoising Based On Optimal Transport](https://mlsanity.com/paper/2312.08135/) (2023)
- [Perceptual Fairness in Image Restoration](https://mlsanity.com/paper/2405.13805/) (2024)
- [Survey on Visual Signal Coding and Processing with Generative Models: Technologies, Standards and Optimization](https://mlsanity.com/paper/2405.14221/) (2024)
- [Rate-Distortion-Perception Theory for the Quadratic Wasserstein Space](https://mlsanity.com/paper/2504.17236/) (2025)
- [DifAugGAN: A Practical Diffusion-style Data Augmentation for GAN-based Single Image Super-resolution](https://mlsanity.com/paper/2311.18508/) (2023)
- [Characterization of the Distortion-Perception Tradeoff for Finite Channels with Arbitrary Metrics](https://mlsanity.com/paper/2402.02265/) (2024)
- [A Simple Combination of Diffusion Models for Better Quality Trade-Offs in Image Denoising](https://mlsanity.com/paper/2503.14654/) (2025)

## Abstract (arXiv)

The lower the distortion of an estimator, the more the distribution of its outputs generally deviates from the distribution of the signals it attempts to estimate. This phenomenon, known as the perception-distortion tradeoff, has captured significant attention in image restoration, where it implies that fidelity to ground truth images comes at the expense of perceptual quality (deviation from statistics of natural images). However, despite the increasing popularity of performing comparisons on the perception-distortion plane, there remains an important open question: what is the minimal distortion that can be achieved under a given perception constraint? In this paper, we derive a closed form expression for this distortion-perception (DP) function for the mean squared-error (MSE) distortion and the Wasserstein-2 perception index. We prove that the DP function is always quadratic, regardless of the underlying distribution. This stems from the fact that estimators on the DP curve form a geodesic in Wasserstein space. In the Gaussian setting, we further provide a closed form expression for such estimators. For general distributions, we show how these estimators can be constructed from the estimators at the two extremes of the tradeoff: The global MSE minimizer, and a minimizer of the MSE under a perfect perceptual quality constraint. The latter can be obtained as a stochastic transformation of the former.