# Revisiting L1 Loss in Super-Resolution: A Probabilistic View and Beyond

arXiv: https://arxiv.org/abs/2201.10084 · HTML: https://mlsanity.com/paper/2201.10084/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 102 references that are in the map.
- First posted on arXiv 25 Jan 2022.

## Builds on (most cited first, 30 of 48)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Perceptual Losses for Real-Time Style Transfer and Super-Resolution](https://mlsanity.com/paper/1603.08155/) (2016)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [SwinIR: Image Restoration Using Swin Transformer](https://mlsanity.com/paper/2108.10257/) (2021)
- [Enhanced Deep Residual Networks for Single Image Super-Resolution](https://mlsanity.com/paper/1707.02921/) (2017)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Image Super-Resolution Using Very Deep Residual Channel Attention Networks](https://mlsanity.com/paper/1807.02758/) (2018)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Accurate Image Super-Resolution Using Very Deep Convolutional Networks](https://mlsanity.com/paper/1511.04587/) (2015)
- [Pre-Trained Image Processing Transformer](https://mlsanity.com/paper/2012.00364/) (2020)
- [Residual Dense Network for Image Super-Resolution](https://mlsanity.com/paper/1802.08797/) (2018)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [FFDNet: Toward a Fast and Flexible Solution for CNN based Image Denoising](https://mlsanity.com/paper/1710.04026/) (2017)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Tutorial on Variational Autoencoders](https://mlsanity.com/paper/1606.05908/) (2016)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [Toward Real-World Single Image Super-Resolution: A New Benchmark and A New Model](https://mlsanity.com/paper/1904.00523/) (2019)
- [Accelerating the Super-Resolution Convolutional Neural Network](https://mlsanity.com/paper/1608.00367/) (2016)
- [Learning Texture Transformer Network for Image Super-Resolution](https://mlsanity.com/paper/2006.04139/) (2020)
- [Sketch-based Manga Retrieval using Manga109 Dataset](https://mlsanity.com/paper/1510.04389/) (2015)
- [Deeply-Recursive Convolutional Network for Image Super-Resolution](https://mlsanity.com/paper/1511.04491/) (2015)
- [Deep multi-scale video prediction beyond mean square error](https://mlsanity.com/paper/1511.05440/) (2015)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [Noise2Self: Blind Denoising by Self-Supervision](https://mlsanity.com/paper/1901.11365/) (2019)
- [Single Image Super-Resolution via a Holistic Attention Network](https://mlsanity.com/paper/2008.08767/) (2020)
- [Fast, Accurate, and Lightweight Super-Resolution with Cascading Residual Network](https://mlsanity.com/paper/1803.08664/) (2018)
- [Residual Non-local Attention Networks for Image Restoration](https://mlsanity.com/paper/1903.10082/) (2019)

## Built on by (most cited first, 5 of 5)

- [TSD-SR: One-Step Diffusion with Target Score Distillation for Real-World Image Super-Resolution](https://mlsanity.com/paper/2411.18263/) (2024)
- [Noise-free Optimization in Early Training Steps for Image Super-Resolution](https://mlsanity.com/paper/2312.17526/) (2023)
- [QUIET-SR: Quantum Image Enhancement Transformer for Single Image Super-Resolution](https://mlsanity.com/paper/2503.08759/) (2025)
- [A Classification-Aware Super-Resolution Framework for Ship Targets in SAR Imagery](https://mlsanity.com/paper/2508.06407/) (2025)
- [Interest Entanglement: The Hidden Barrier to Blind Super-Resolution Optimization](https://mlsanity.com/paper/2606.22353/) (2026)

## Abstract (arXiv)

Super-resolution as an ill-posed problem has many high-resolution candidates for a low-resolution input. However, the popular ℓ1 loss used to best fit the given HR image fails to consider this fundamental property of non-uniqueness in image restoration. In this work, we fix the missing piece in ℓ1 loss by formulating super-resolution with neural networks as a probabilistic model. It shows that ℓ1 loss is equivalent to a degraded likelihood function that removes the randomness from the learning process. By introducing a data-adaptive random variable, we present a new objective function that aims at minimizing the expectation of the reconstruction error over all plausible solutions. The experimental results show consistent improvements on mainstream architectures, with no extra parameter or computing cost at inference time.