# A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing

arXiv: https://arxiv.org/abs/2310.03758 · HTML: https://mlsanity.com/paper/2310.03758/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 52 references that are in the map.
- First posted on arXiv 25 Sep 2023.

## Builds on (most cited first, 27 of 27)

- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Invertible generative models for inverse problems: mitigating representation error and dataset bias](https://mlsanity.com/paper/1905.11672/) (2019)
- [Compressed Sensing MRI Reconstruction using a Generative Adversarial Network with a Cyclic Loss](https://mlsanity.com/paper/1709.00753/) (2017)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [Intermediate Layer Optimization for Inverse Problems using Deep Generative Models](https://mlsanity.com/paper/2102.07364/) (2021)
- [Composing Normalizing Flows for Inverse Problems](https://mlsanity.com/paper/2002.11743/) (2020)
- [Instance-Optimal Compressed Sensing via Posterior Sampling](https://mlsanity.com/paper/2106.11438/) (2021)
- [Global Guarantees for Enforcing Deep Generative Priors by Empirical Risk](https://mlsanity.com/paper/1705.07576/) (2017)
- [Modeling Sparse Deviations for Compressed Sensing using Generative Models](https://mlsanity.com/paper/1807.01442/) (2018)
- [Robust Compressed Sensing using Generative Models](https://mlsanity.com/paper/2006.09461/) (2020)
- [Information-Theoretic Lower Bounds for Compressive Sensing with Generative Models](https://mlsanity.com/paper/1908.10744/) (2019)
- [Theoretical Perspectives on Deep Learning Methods in Inverse Problems](https://mlsanity.com/paper/2206.14373/) (2022)
- [Sample Complexity Bounds for 1-bit Compressive Sensing and Binary Stable Embeddings with Generative Priors](https://mlsanity.com/paper/2002.01697/) (2020)
- [The Generalized Lasso with Nonlinear Observations and Generative Priors](https://mlsanity.com/paper/2006.12415/) (2020)
- [Quantized Compressed Sensing with Score-based Generative Models](https://mlsanity.com/paper/2211.13006/) (2022)
- [Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors](https://mlsanity.com/paper/2106.15358/) (2021)
- [High Dimensional Statistical Estimation under Uniformly Dithered One-bit Quantization](https://mlsanity.com/paper/2202.13157/) (2022)
- [The Generalized Lasso for Sub-gaussian Measurements with Dithered Quantization](https://mlsanity.com/paper/1807.06976/) (2018)
- [Non-Iterative Recovery from Nonlinear Observations using Generative Models](https://mlsanity.com/paper/2205.15749/) (2022)
- [Quantizing Heavy-tailed Data in Statistical Estimation: (Near) Minimax Rates, Covariate Quantization, and Uniform Recovery](https://mlsanity.com/paper/2212.14562/) (2022)
- [Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices](https://mlsanity.com/paper/1306.1154/) (2013)
- [Projected Gradient Descent Algorithms for Solving Nonlinear Inverse Problems with Generative Priors](https://mlsanity.com/paper/2209.10093/) (2022)
- [Just Least Squares: Binary Compressive Sampling with Low Generative Intrinsic Dimension](https://mlsanity.com/paper/2111.14486/) (2021)

## Built on by (most cited first, 7 of 7)

- [Integrating Intermediate Layer Optimization and Projected Gradient Descent for Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2505.20789/) (2025)
- [A unified framework for learning with nonlinear model classes from arbitrary linear samples](https://mlsanity.com/paper/2311.14886/) (2023)
- [Generalized Eigenvalue Problems with Generative Priors](https://mlsanity.com/paper/2411.01326/) (2024)
- [Learning Single Index Models with Diffusion Priors](https://mlsanity.com/paper/2505.21135/) (2025)
- [Diffusion Model Based Signal Recovery Under 1-Bit Quantization](https://mlsanity.com/paper/2511.12471/) (2025)
- [Active Learning for Conditional Generative Compressed Sensing](https://mlsanity.com/paper/2605.05435/) (2026)
- [Tracing the Oracle: Improving Diffusion Timestep Scheduling for 3D CT Reconstruction](https://mlsanity.com/paper/2606.06236/) (2026)

## Abstract (arXiv)

In generative compressed sensing (GCS), we want to recover a signal x^* ∈ ℝn from m measurements (mll n) using a generative prior x^*∈ G(mathbbB2k(r)), where G is typically an L-Lipschitz continuous generative model and mathbbB2k(r) represents the radius-r ℓ2-ball in ℝk. Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed x^* rather than for all x^* simultaneously. In this paper, we build a unified framework to derive uniform recovery guarantees for nonlinear GCS where the observation model is nonlinear and possibly discontinuous or unknown. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. Specifically, using a single realization of the sensing ensemble and generalized Lasso, em all x^*∈ G(mathbbB2k(r)) can be recovered up to an ℓ2-error at most ε using roughly tildeO(k/ε2) samples, with omitted logarithmic factors typically being dominated by log L. Notably, this almost coincides with existing non-uniform guarantees up to logarithmic factors, hence the uniformity costs very little. As part of our technical contributions, we introduce the Lipschitz approximation to handle discontinuous observation models. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy. Experimental results are presented to corroborate our theory.