# Projected Gradient Descent Algorithms for Solving Nonlinear Inverse Problems with Generative Priors

arXiv: https://arxiv.org/abs/2209.10093 · HTML: https://mlsanity.com/paper/2209.10093/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 33 references that are in the map.
- First posted on arXiv 21 Sep 2022.

## Builds on (most cited first, 18 of 18)

- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [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)
- [Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks](https://mlsanity.com/paper/1810.03982/) (2018)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [Solving Linear Inverse Problems Using GAN Priors: An Algorithm with Provable Guarantees](https://mlsanity.com/paper/1802.08406/) (2018)
- [Instance-Optimal Compressed Sensing via Posterior Sampling](https://mlsanity.com/paper/2106.11438/) (2021)
- [Algorithmic Guarantees for Inverse Imaging with Untrained Network Priors](https://mlsanity.com/paper/1906.08763/) (2019)
- [Information-Theoretic Lower Bounds for Compressive Sensing with Generative Models](https://mlsanity.com/paper/1908.10744/) (2019)
- [Sample Complexity Bounds for 1-bit Compressive Sensing and Binary Stable Embeddings with Generative Priors](https://mlsanity.com/paper/2002.01697/) (2020)
- [Alternating Phase Projected Gradient Descent with Generative Priors for Solving Compressive Phase Retrieval](https://mlsanity.com/paper/1903.02707/) (2019)
- [The Generalized Lasso with Nonlinear Observations and Generative Priors](https://mlsanity.com/paper/2006.12415/) (2020)
- [Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors](https://mlsanity.com/paper/2106.15358/) (2021)
- [Generative Principal Component Analysis](https://mlsanity.com/paper/2203.09693/) (2022)
- [Robust One-Bit Recovery via ReLU Generative Networks: Near-Optimal Statistical Rate and Global Landscape Analysis](https://mlsanity.com/paper/1908.05368/) (2019)
- [Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors](https://mlsanity.com/paper/2108.03570/) (2021)
- [Fast and Reliable Parameter Estimation from Nonlinear Observations](https://mlsanity.com/paper/1610.07108/) (2016)
- [Fast Algorithms for Demixing Sparse Signals from Nonlinear Observations](https://mlsanity.com/paper/1608.01234/) (2016)

## Built on by (most cited first, 5 of 5)

- [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)
- [Image Restoration via Diffusion Models with Dynamic Resolution](https://mlsanity.com/paper/2605.14267/) (2026)
- [Diffusion Model Based Signal Recovery Under 1-Bit Quantization](https://mlsanity.com/paper/2511.12471/) (2025)

## Abstract (arXiv)

In this paper, we propose projected gradient descent (PGD) algorithms for signal estimation from noisy nonlinear measurements. We assume that the unknown p-dimensional signal lies near the range of an L-Lipschitz continuous generative model with bounded k-dimensional inputs. In particular, we consider two cases when the nonlinear link function is either unknown or known. For unknown nonlinearity, similarly to citeliu2020generalized, we make the assumption of sub-Gaussian observations and propose a linear least-squares estimator. We show that when there is no representation error and the sensing vectors are Gaussian, roughly O(k log L) samples suffice to ensure that a PGD algorithm converges linearly to a point achieving the optimal statistical rate using arbitrary initialization. For known nonlinearity, we assume monotonicity as in citeyang2016sparse, and make much weaker assumptions on the sensing vectors and allow for representation error. We propose a nonlinear least-squares estimator that is guaranteed to enjoy an optimal statistical rate. A corresponding PGD algorithm is provided and is shown to also converge linearly to the estimator using arbitrary initialization. In addition, we present experimental results on image datasets to demonstrate the performance of our PGD algorithms.