# Sensing Theorems for Unsupervised Learning in Linear Inverse Problems

arXiv: https://arxiv.org/abs/2203.12513 · HTML: https://mlsanity.com/paper/2203.12513/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 54 references that are in the map.
- First posted on arXiv 23 Mar 2022.

## Builds on (most cited first, 16 of 16)

- [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)
- [Learned Initializations for Optimizing Coordinate-Based Neural Representations](https://mlsanity.com/paper/2012.02189/) (2020)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Equivariant Imaging: Learning Beyond the Range Space](https://mlsanity.com/paper/2103.14756/) (2021)
- [Sparse and Dense Data with CNNs: Depth Completion and Semantic Segmentation](https://mlsanity.com/paper/1808.00769/) (2018)
- [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)
- [Unsupervised Learning From Incomplete Measurements for Inverse Problems](https://mlsanity.com/paper/2201.12151/) (2022)
- [A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion](https://mlsanity.com/paper/1503.02596/) (2015)
- [Algebraic Variety Models for High-Rank Matrix Completion](https://mlsanity.com/paper/1703.09631/) (2017)
- [Compressive Statistical Learning with Random Feature Moments](https://mlsanity.com/paper/1706.07180/) (2017)
- [Fundamental Limits in Multi-image Alignment](https://mlsanity.com/paper/1602.01541/) (2016)

## Built on by (most cited first, 28 of 28)

- [Equivariant Multi-Modality Image Fusion](https://mlsanity.com/paper/2305.11443/) (2023)
- [Imaging with Equivariant Deep Learning](https://mlsanity.com/paper/2209.01725/) (2022)
- [Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data](https://mlsanity.com/paper/2403.08728/) (2024)
- [Equivariant plug-and-play image reconstruction](https://mlsanity.com/paper/2312.01831/) (2023)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Scale-Equivariant Imaging: Self-Supervised Learning for Image Super-Resolution and Deblurring](https://mlsanity.com/paper/2312.11232/) (2023)
- [Self-Supervised Scalable Deep Compressed Sensing](https://mlsanity.com/paper/2308.13777/) (2023)
- [Perspective-Equivariance for Unsupervised Imaging with Camera Geometry](https://mlsanity.com/paper/2403.09327/) (2024)
- [Equivariance-based self-supervised learning for audio signal recovery from clipped measurements](https://mlsanity.com/paper/2409.15283/) (2024)
- [Self-Supervised Learning from Noisy and Incomplete Data](https://mlsanity.com/paper/2601.03244/) (2026)
- [Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging](https://mlsanity.com/paper/2311.08972/) (2023)
- [Unsupervised Detection of Distribution Shift in Inverse Problems using Diffusion Models](https://mlsanity.com/paper/2505.11482/) (2025)
- [The Star Geometry of Critic-Based Regularizer Learning](https://mlsanity.com/paper/2408.16852/) (2024)
- [EquiReg: Equivariance Regularized Diffusion for Inverse Problems](https://mlsanity.com/paper/2505.22973/) (2025)
- [Normalization-equivariant Diffusion Models: Learning Posterior Samplers From Noisy And Partial Measurements](https://mlsanity.com/paper/2510.11964/) (2025)
- [Efficient Model-Based Deep Learning via Network Pruning and Fine-Tuning](https://mlsanity.com/paper/2311.02003/) (2023)
- [Reconstruct Anything Model: a lightweight general model for computational imaging](https://mlsanity.com/paper/2503.08915/) (2025)
- [Double Blind Imaging with Generative Modeling](https://mlsanity.com/paper/2503.21501/) (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)
- [SHARE: A Fully Unsupervised Framework for Single Hyperspectral Image Restoration](https://mlsanity.com/paper/2601.13987/) (2026)
- [Equivariant Test-Time Training with Operator Sketching for Imaging Inverse Problems](https://mlsanity.com/paper/2411.05771/) (2024)
- [Training-free Ultra Small Model for Universal Sparse Reconstruction in Compressed Sensing](https://mlsanity.com/paper/2501.11592/) (2025)
- [Self-supervised learning for phase retrieval](https://mlsanity.com/paper/2509.26203/) (2025)
- [Equivariant Learning for Unsupervised Image Dehazing](https://mlsanity.com/paper/2601.13986/) (2026)
- [Scale Equivariance Regularization and Feature Lifting in High Dynamic Range Modulo Imaging](https://mlsanity.com/paper/2601.23037/) (2026)
- [Learning to reconstruct from saturated data: audio declipping and high-dynamic range imaging](https://mlsanity.com/paper/2602.22279/) (2026)
- [Priors learned from legacy reconstructions inherit undetectable overconfidence](https://mlsanity.com/paper/2607.21721/) (2026)

## Abstract (arXiv)

Solving an ill-posed linear inverse problem requires knowledge about the underlying signal model. In many applications, this model is a priori unknown and has to be learned from data. However, it is impossible to learn the model using observations obtained via a single incomplete measurement operator, as there is no information about the signal model in the nullspace of the operator, resulting in a chicken-and-egg problem: to learn the model we need reconstructed signals, but to reconstruct the signals we need to know the model. Two ways to overcome this limitation are using multiple measurement operators or assuming that the signal model is invariant to a certain group action. In this paper, we present necessary and sufficient sensing conditions for learning the signal model from measurement data alone which only depend on the dimension of the model and the number of operators or properties of the group action that the model is invariant to. As our results are agnostic of the learning algorithm, they shed light into the fundamental limitations of learning from incomplete data and have implications in a wide range set of practical algorithms, such as dictionary learning, matrix completion and deep neural networks.