# Unsupervised Learning with Stein's Unbiased Risk Estimator

arXiv: https://arxiv.org/abs/1805.10531 · HTML: https://mlsanity.com/paper/1805.10531/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 May 2018.

## Built on by (most cited first, 30 of 47)

- [Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurements](https://mlsanity.com/paper/2111.12855/) (2021)
- [DDM2: Self-Supervised Diffusion MRI Denoising with Generative Diffusion Models](https://mlsanity.com/paper/2302.03018/) (2023)
- [Early Stopping for Deep Image Prior](https://mlsanity.com/paper/2112.06074/) (2021)
- [Consistent Diffusion Meets Tweedie: Training Exact Ambient Diffusion Models with Noisy Data](https://mlsanity.com/paper/2404.10177/) (2024)
- [Solving Inverse Problems with Score-Based Generative Priors learned from Noisy Data](https://mlsanity.com/paper/2305.01166/) (2023)
- [GSURE-Based Diffusion Model Training with Corrupted Data](https://mlsanity.com/paper/2305.13128/) (2023)
- [Rethinking Deep Image Prior for Denoising](https://mlsanity.com/paper/2108.12841/) (2021)
- [Sensing Theorems for Unsupervised Learning in Linear Inverse Problems](https://mlsanity.com/paper/2203.12513/) (2022)
- [FBI-Denoiser: Fast Blind Image Denoiser for Poisson-Gaussian Noise](https://mlsanity.com/paper/2105.10967/) (2021)
- [Unsupervised Learning From Incomplete Measurements for Inverse Problems](https://mlsanity.com/paper/2201.12151/) (2022)
- [Adaptive Denoising via GainTuning](https://mlsanity.com/paper/2107.12815/) (2021)
- [Scale-Equivariant Imaging: Self-Supervised Learning for Image Super-Resolution and Deblurring](https://mlsanity.com/paper/2312.11232/) (2023)
- [A review and experimental evaluation of deep learning methods for MRI reconstruction](https://mlsanity.com/paper/2109.08618/) (2021)
- [Deep Internal Learning: Deep Learning from a Single Input](https://mlsanity.com/paper/2312.07425/) (2023)
- [Deep Learning for Accelerated and Robust MRI Reconstruction: a Review](https://mlsanity.com/paper/2404.15692/) (2024)
- [Image Restoration by Deep Projected GSURE](https://mlsanity.com/paper/2102.02485/) (2021)
- [Self-Supervised Scalable Deep Compressed Sensing](https://mlsanity.com/paper/2308.13777/) (2023)
- [Thompson Sampling with Diffusion Generative Prior](https://mlsanity.com/paper/2301.05182/) (2023)
- [A Comparison of Image Denoising Methods](https://mlsanity.com/paper/2304.08990/) (2023)
- [K-band: Self-supervised MRI Reconstruction via Stochastic Gradient Descent over K-space Subsets](https://mlsanity.com/paper/2308.02958/) (2023)
- [Model Adaptation for Image Reconstruction using Generalized Stein's Unbiased Risk Estimator](https://mlsanity.com/paper/2102.00047/) (2021)
- [Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging](https://mlsanity.com/paper/2311.08972/) (2023)
- [Robust multi-coil MRI reconstruction via self-supervised denoising](https://mlsanity.com/paper/2411.12919/) (2024)
- [Benchmarking Self-Supervised Learning Methods for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2502.14009/) (2025)
- [Evaluating Unsupervised Denoising Requires Unsupervised Metrics](https://mlsanity.com/paper/2210.05553/) (2022)
- [Unsupervised Training of Convex Regularizers using Maximum Likelihood Estimation](https://mlsanity.com/paper/2404.05445/) (2024)
- [DiffDenoise: Self-Supervised Medical Image Denoising with Conditional Diffusion Models](https://mlsanity.com/paper/2504.00264/) (2025)
- [Normalization-equivariant Diffusion Models: Learning Posterior Samplers From Noisy And Partial Measurements](https://mlsanity.com/paper/2510.11964/) (2025)
- [CDLNet: Noise-Adaptive Convolutional Dictionary Learning Network for Blind Denoising and Demosaicing](https://mlsanity.com/paper/2112.00913/) (2021)
- [SVD-DIP: Overcoming the Overfitting Problem in DIP-based CT Reconstruction](https://mlsanity.com/paper/2303.15748/) (2023)

## Abstract (arXiv)

Learning from unlabeled and noisy data is one of the grand challenges of machine learning. As such, it has seen a flurry of research with new ideas proposed continuously. In this work, we revisit a classical idea: Stein's Unbiased Risk Estimator (SURE). We show that, in the context of image recovery, SURE and its generalizations can be used to train convolutional neural networks (CNNs) for a range of image denoising and recovery problems without any ground truth data. Specifically, our goal is to reconstruct an image x from a noisy linear transformation (measurement) of the image. We consider two scenarios: one where no additional data is available and one where we have measurements of other images that are drawn from the same noisy distribution as x, but have no access to the clean images. Such is the case, for instance, in the context of medical imaging, microscopy, and astronomy, where noise-less ground truth data is rarely available. We show that in this situation, SURE can be used to estimate the mean-squared-error loss associated with an estimate of x. Using this estimate of the loss, we train networks to perform denoising and compressed sensing recovery. In addition, we also use the SURE framework to partially explain and improve upon an intriguing results presented by Ulyanov et al. in "Deep Image Prior": that a network initialized with random weights and fit to a single noisy image can effectively denoise that image. Public implementations of the networks and methods described in this paper can be found at https://github.com/ricedsp/D-AMPToolbox.