# Modeling Sparse Deviations for Compressed Sensing using Generative Models

arXiv: https://arxiv.org/abs/1807.01442 · HTML: https://mlsanity.com/paper/1807.01442/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jul 2018.

## Built on by (most cited first, 30 of 33)

- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Intermediate Layer Optimization for Inverse Problems using Deep Generative Models](https://mlsanity.com/paper/2102.07364/) (2021)
- [Instance-Optimal Compressed Sensing via Posterior Sampling](https://mlsanity.com/paper/2106.11438/) (2021)
- [A Theoretical Justification for Image Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2302.01217/) (2023)
- [Regularising Inverse Problems with Generative Machine Learning Models](https://mlsanity.com/paper/2107.11191/) (2021)
- [Theoretical Perspectives on Deep Learning Methods in Inverse Problems](https://mlsanity.com/paper/2206.14373/) (2022)
- [ADIR: Adaptive Diffusion for Image Reconstruction](https://mlsanity.com/paper/2212.03221/) (2022)
- [Generative Principal Component Analysis](https://mlsanity.com/paper/2203.09693/) (2022)
- [Score-Guided Intermediate Layer Optimization: Fast Langevin Mixing for Inverse Problems](https://mlsanity.com/paper/2206.09104/) (2022)
- [Non-Iterative Recovery from Nonlinear Observations using Generative Models](https://mlsanity.com/paper/2205.15749/) (2022)
- [Inverse Problems Leveraging Pre-trained Contrastive Representations](https://mlsanity.com/paper/2110.07439/) (2021)
- [Misspecified Phase Retrieval with Generative Priors](https://mlsanity.com/paper/2210.05571/) (2022)
- [A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing](https://mlsanity.com/paper/2310.03758/) (2023)
- [Integrating Intermediate Layer Optimization and Projected Gradient Descent for Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2505.20789/) (2025)
- [Assessment of the Reliablity of a Model's Decision by Generalizing Attribution to the Wavelet Domain](https://mlsanity.com/paper/2305.14979/) (2023)
- [Compressed Sensing MRI Reconstruction Regularized by VAEs with Structured Image Covariance](https://mlsanity.com/paper/2210.14586/) (2022)
- [Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priors](https://mlsanity.com/paper/2309.09032/) (2023)
- [Variational Bayes image restoration with compressive autoencoders](https://mlsanity.com/paper/2311.17744/) (2023)
- [Injectivity capacity of ReLU gates](https://mlsanity.com/paper/2410.20646/) (2024)
- [How many measurements are enough? Bayesian recovery in inverse problems with general distributions](https://mlsanity.com/paper/2505.10630/) (2025)
- [Learning Single Index Models with Diffusion Priors](https://mlsanity.com/paper/2505.21135/) (2025)
- [Non-Convex Compressed Sensing with Training Data](https://mlsanity.com/paper/2101.08310/) (2021)
- [Provably Convergent Algorithms for Solving Inverse Problems Using Generative Models](https://mlsanity.com/paper/2105.06371/) (2021)
- [Learning Generative Prior with Latent Space Sparsity Constraints](https://mlsanity.com/paper/2105.11956/) (2021)
- [GenMod: A generative modeling approach for spectral representation of PDEs with random inputs](https://mlsanity.com/paper/2201.12973/) (2022)
- [Regularized Training of Intermediate Layers for Generative Models for Inverse Problems](https://mlsanity.com/paper/2203.04382/) (2022)
- [Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview](https://mlsanity.com/paper/2402.12072/) (2024)
- [Deep priors for satellite image restoration with accurate uncertainties](https://mlsanity.com/paper/2412.04130/) (2024)
- [A Diffusion-Based Generative Prior Approach to Sparse-view Computed Tomography](https://mlsanity.com/paper/2602.10722/) (2026)
- [Learning Trees of ℓ0-Minimization Problems](https://mlsanity.com/paper/2302.02548/) (2023)

## Abstract (arXiv)

In compressed sensing, a small number of linear measurements can be used to reconstruct an unknown signal. Existing approaches leverage assumptions on the structure of these signals, such as sparsity or the availability of a generative model. A domain-specific generative model can provide a stronger prior and thus allow for recovery with far fewer measurements. However, unlike sparsity-based approaches, existing methods based on generative models guarantee exact recovery only over their support, which is typically only a small subset of the space on which the signals are defined. We propose Sparse-Gen, a framework that allows for sparse deviations from the support set, thereby achieving the best of both worlds by using a domain specific prior and allowing reconstruction over the full space of signals. Theoretically, our framework provides a new class of signals that can be acquired using compressed sensing, reducing classic sparse vector recovery to a special case and avoiding the restrictive support due to a generative model prior. Empirically, we observe consistent improvements in reconstruction accuracy over competing approaches, especially in the more practical setting of transfer compressed sensing where a generative model for a data-rich, source domain aids sensing on a data-scarce, target domain.