mlsanityData to 7 Oct 2026

Sensing Theorems for Unsupervised Learning in Linear Inverse Problems

arXiv 2203.12513 ↗First posted 23 Mar 2022stat.ML cs.LG eess.IVcited by 28

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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.

Key facts. Cited by 28 papers in the map (data through 7 Oct 2026). Builds on 16 of its 54 references that are in the map. First posted on arXiv 23 Mar 2022.

Builds on

16 of 54 references.

References · 16

  1. 2017Deep Image Priorcited by 1,097
  2. 2016Deep Convolutional Neural Network for Inverse Problems in Imagingcited by 431
  3. 2019Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processingcited by 411
  4. 2020Learned Initializations for Optimizing Coordinate-Based Neural Representationscited by 240
  5. 2019Deep Scale-spaces: Equivariance Over Scalecited by 113
  6. 2021Equivariant Imaging: Learning Beyond the Range Spacecited by 83
  7. 2018Sparse and Dense Data with CNNs: Depth Completion and Semantic Segmentationcited by 76
  8. 2019RARE: Image Reconstruction using Deep Priors Learned without Ground Truthcited by 64
  9. 2021Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurementscited by 56
  10. 2018Unsupervised Learning with Stein's Unbiased Risk Estimatorcited by 47
  11. 2013Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programmingcited by 20
  12. 2022Unsupervised Learning From Incomplete Measurements for Inverse Problemscited by 20
4 more
  1. 2015A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completioncited by 13
  2. 2017Algebraic Variety Models for High-Rank Matrix Completioncited by 11
  3. 2017Compressive Statistical Learning with Random Feature Momentscited by 11
  4. 2016Fundamental Limits in Multi-image Alignmentcited by 6

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 7; 2024: 5; 2025: 9; 2026: 6.

28 papers.

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  1. 2023Equivariant Multi-Modality Image Fusioncited by 98
  2. 2022Imaging with Equivariant Deep Learningcited by 35
  3. 2024Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Datacited by 32
  4. 2023Equivariant plug-and-play image reconstructioncited by 26
  5. 2023What's in a Prior? Learned Proximal Networks for Inverse Problemscited by 24
  6. 2023Scale-Equivariant Imaging: Self-Supervised Learning for Image Super-Resolution and Deblurringcited by 14
  7. 2023Self-Supervised Scalable Deep Compressed Sensingcited by 8
  8. 2024Perspective-Equivariance for Unsupervised Imaging with Camera Geometrycited by 5
  9. 2024Equivariance-based self-supervised learning for audio signal recovery from clipped measurementscited by 5
  10. 2026Self-Supervised Learning from Noisy and Incomplete Datacited by 5
  11. 2023Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imagingcited by 4
  12. 2025Unsupervised Detection of Distribution Shift in Inverse Problems using Diffusion Modelscited by 4
16 more
  1. 2024The Star Geometry of Critic-Based Regularizer Learningcited by 3
  2. 2025Normalization-equivariant Diffusion Models: Learning Posterior Samplers From Noisy And Partial Measurementscited by 3
  3. 2023Efficient Model-Based Deep Learning via Network Pruning and Fine-Tuningcited by 2
  4. 2025Reconstruct Anything Model: a lightweight general model for computational imagingcited by 1
  5. 2025Double Blind Imaging with Generative Modelingcited by 1
  6. 2025EquiReg: Equivariance Regularized Diffusion for Inverse Problemscited by 1
  7. 2025Equivariant Splitting: Self-supervised learning from incomplete datacited by 1
  8. 2026SHARE: A Fully Unsupervised Framework for Single Hyperspectral Image Restorationcited by 1
  9. 2024Equivariant Test-Time Training with Operator Sketching for Imaging Inverse Problemscited by 0
  10. 2025Training-free Ultra Small Model for Universal Sparse Reconstruction in Compressed Sensingcited by 0
  11. 2025Self-supervised learning for phase retrievalcited by 0
  12. 2025UPMRI: Unsupervised Parallel MRI Reconstruction via Projected Conditional Flow Matchingcited by 0
  13. 2026Equivariant Learning for Unsupervised Image Dehazingcited by 0
  14. 2026Scale Equivariance Regularization and Feature Lifting in High Dynamic Range Modulo Imagingcited by 0
  15. 2026Learning to reconstruct from saturated data: audio declipping and high-dynamic range imagingcited by 0
  16. 2026Priors learned from legacy reconstructions inherit undetectable overconfidencecited by 0