# Unsupervised Learning From Incomplete Measurements for Inverse Problems

arXiv: https://arxiv.org/abs/2201.12151 · HTML: https://mlsanity.com/paper/2201.12151/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 53 references that are in the map.
- First posted on arXiv 28 Jan 2022.

## Builds on (most cited first, 15 of 15)

- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Equivariant Imaging: Learning Beyond the Range Space](https://mlsanity.com/paper/2103.14756/) (2021)
- [RARE: Image Reconstruction using Deep Priors Learned without Ground Truth](https://mlsanity.com/paper/1912.05854/) (2019)
- [Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurements](https://mlsanity.com/paper/2111.12855/) (2021)
- [Unsupervised Learning with Stein's Unbiased Risk Estimator](https://mlsanity.com/paper/1805.10531/) (2018)
- [Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming](https://mlsanity.com/paper/1310.0807/) (2013)
- [Training deep learning based image denoisers from undersampled measurements without ground truth and without image prior](https://mlsanity.com/paper/1806.00961/) (2018)
- [A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion](https://mlsanity.com/paper/1503.02596/) (2015)
- [Training Image Estimators without Image Ground-Truth](https://mlsanity.com/paper/1906.05775/) (2019)

## Built on by (most cited first, 20 of 20)

- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data](https://mlsanity.com/paper/2403.08728/) (2024)
- [Sensing Theorems for Unsupervised Learning in Linear Inverse Problems](https://mlsanity.com/paper/2203.12513/) (2022)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Theoretical Perspectives on Deep Learning Methods in Inverse Problems](https://mlsanity.com/paper/2206.14373/) (2022)
- [Generalized Recorrupted-to-Recorrupted: Self-Supervised Learning Beyond Gaussian Noise](https://mlsanity.com/paper/2412.04648/) (2024)
- [Self-Supervised Scalable Deep Compressed Sensing](https://mlsanity.com/paper/2308.13777/) (2023)
- [Learning to Reconstruct Signals From Binary Measurements](https://mlsanity.com/paper/2303.08691/) (2023)
- [Equivariant Imaging for Self-supervised Hyperspectral Image Inpainting](https://mlsanity.com/paper/2404.13159/) (2024)
- [Self-Supervised Learning from Noisy and Incomplete Data](https://mlsanity.com/paper/2601.03244/) (2026)
- [Benchmarking Self-Supervised Learning Methods for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2502.14009/) (2025)
- [Sparsity-Driven Parallel Imaging Consistency for Improved Self-Supervised MRI Reconstruction](https://mlsanity.com/paper/2505.24136/) (2025)
- [The Star Geometry of Critic-Based Regularizer Learning](https://mlsanity.com/paper/2408.16852/) (2024)
- [Normalization-equivariant Diffusion Models: Learning Posterior Samplers From Noisy And Partial Measurements](https://mlsanity.com/paper/2510.11964/) (2025)
- [Reconstruct Anything Model: a lightweight general model for computational imaging](https://mlsanity.com/paper/2503.08915/) (2025)
- [Equivariant Splitting: Self-supervised learning from incomplete data](https://mlsanity.com/paper/2510.00929/) (2025)
- [UPMRI: Unsupervised Parallel MRI Reconstruction via Projected Conditional Flow Matching](https://mlsanity.com/paper/2512.17493/) (2025)
- [Training-Free Adversarial Robustness in Computational MRI](https://mlsanity.com/paper/2501.01908/) (2025)
- [Fast Equivariant Imaging: Accelerating Unsupervised Learning and Model Adaptation via Inexact Splitting](https://mlsanity.com/paper/2507.06764/) (2025)
- [Frozen CLIP Priors for Robust Self-Supervised Poisson Inverse Problems](https://mlsanity.com/paper/2608.20524/) (2026)

## Abstract (arXiv)

In many real-world inverse problems, only incomplete measurement data are available for training which can pose a problem for learning a reconstruction function. Indeed, unsupervised learning using a fixed incomplete measurement process is impossible in general, as there is no information in the nullspace of the measurement operator. This limitation can be overcome by using measurements from multiple operators. While this idea has been successfully applied in various applications, a precise characterization of the conditions for learning is still lacking. In this paper, we fill this gap by presenting necessary and sufficient conditions for learning the underlying signal model needed for reconstruction which indicate the interplay between the number of distinct measurement operators, the number of measurements per operator, the dimension of the model and the dimension of the signals. Furthermore, we propose a novel and conceptually simple unsupervised learning loss which only requires access to incomplete measurement data and achieves a performance on par with supervised learning when the sufficient condition is verified. We validate our theoretical bounds and demonstrate the advantages of the proposed unsupervised loss compared to previous methods via a series of experiments on various imaging inverse problems, such as accelerated magnetic resonance imaging, compressed sensing and image inpainting.