# Misspecified Phase Retrieval with Generative Priors

arXiv: https://arxiv.org/abs/2210.05571 · HTML: https://mlsanity.com/paper/2210.05571/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 103 references that are in the map.
- First posted on arXiv 11 Oct 2022.

## Builds on (most cited first, 30 of 48)

- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [PULSE: Self-Supervised Photo Upsampling via Latent Space Exploration of Generative Models](https://mlsanity.com/paper/2003.03808/) (2020)
- [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)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks](https://mlsanity.com/paper/1810.03982/) (2018)
- [Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models](https://mlsanity.com/paper/1708.03395/) (2017)
- [Invertible generative models for inverse problems: mitigating representation error and dataset bias](https://mlsanity.com/paper/1905.11672/) (2019)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval](https://mlsanity.com/paper/1803.07726/) (2018)
- [One Network to Solve Them All --- Solving Linear Inverse Problems using Deep Projection Models](https://mlsanity.com/paper/1703.09912/) (2017)
- [Phase Retrieval using Alternating Minimization](https://mlsanity.com/paper/1306.0160/) (2013)
- [Solving Linear Inverse Problems Using GAN Priors: An Algorithm with Provable Guarantees](https://mlsanity.com/paper/1802.08406/) (2018)
- [Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems](https://mlsanity.com/paper/1505.05114/) (2015)
- [Compressed Sensing with Deep Image Prior and Learned Regularization](https://mlsanity.com/paper/1806.06438/) (2018)
- [Deep Compressed Sensing](https://mlsanity.com/paper/1905.06723/) (2019)
- [Instance-Optimal Compressed Sensing via Posterior Sampling](https://mlsanity.com/paper/2106.11438/) (2021)
- [Fundamental Limits of Weak Recovery with Applications to Phase Retrieval](https://mlsanity.com/paper/1708.05932/) (2017)
- [Phase retrieval in high dimensions: Statistical and computational phase transitions](https://mlsanity.com/paper/2006.05228/) (2020)
- [Global Guarantees for Enforcing Deep Generative Priors by Empirical Risk](https://mlsanity.com/paper/1705.07576/) (2017)
- [Bootstrapping Upper Confidence Bound](https://mlsanity.com/paper/1906.05247/) (2019)
- [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)
- [Algorithmic Guarantees for Inverse Imaging with Untrained Network Priors](https://mlsanity.com/paper/1906.08763/) (2019)
- [Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow](https://mlsanity.com/paper/1605.08285/) (2016)
- [Phase Transitions of Spectral Initialization for High-Dimensional Nonconvex Estimation](https://mlsanity.com/paper/1702.06435/) (2017)
- [PhaseMax: Convex Phase Retrieval via Basis Pursuit](https://mlsanity.com/paper/1610.07531/) (2016)
- [Information-Theoretic Lower Bounds for Compressive Sensing with Generative Models](https://mlsanity.com/paper/1908.10744/) (2019)
- [Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow](https://mlsanity.com/paper/1506.03382/) (2015)

## 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)
- [Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priors](https://mlsanity.com/paper/2309.09032/) (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)
- [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)
- [Tracing the Oracle: Improving Diffusion Timestep Scheduling for 3D CT Reconstruction](https://mlsanity.com/paper/2606.06236/) (2026)

## Abstract (arXiv)

In this paper, we study phase retrieval under model misspecification and generative priors. In particular, we aim to estimate an n-dimensional signal x from m i.i.d.~realizations of the single index model y = f(aTx), where f is an unknown and possibly random nonlinear link function and a ∈ ℝn is a standard Gaussian vector. We make the assumption Cov[y,(aTx)2] ne 0, which corresponds to the misspecified phase retrieval problem. In addition, the underlying signal x is assumed to lie in the range of an L-Lipschitz continuous generative model with bounded k-dimensional inputs. We propose a two-step approach, for which the first step plays the role of spectral initialization and the second step refines the estimated vector produced by the first step iteratively. We show that both steps enjoy a statistical rate of order √(klog L)· (log m)/m under suitable conditions. Experiments on image datasets are performed to demonstrate that our approach performs on par with or even significantly outperforms several competing methods.