# Phase Retrieval Under a Generative Prior

arXiv: https://arxiv.org/abs/1807.04261 · HTML: https://mlsanity.com/paper/1807.04261/

## Key facts

- Cited by 73 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2018.

## Built on by (most cited first, 30 of 73)

- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [GAN Prior Embedded Network for Blind Face Restoration in the Wild](https://mlsanity.com/paper/2105.06070/) (2021)
- [Intermediate Layer Optimization for Inverse Problems using Deep Generative Models](https://mlsanity.com/paper/2102.07364/) (2021)
- [Early Stopping for Deep Image Prior](https://mlsanity.com/paper/2112.06074/) (2021)
- [Instance-Optimal Compressed Sensing via Posterior Sampling](https://mlsanity.com/paper/2106.11438/) (2021)
- [Learning to Solve PDE-constrained Inverse Problems with Graph Networks](https://mlsanity.com/paper/2206.00711/) (2022)
- [Deterministic equivalent and error universality of deep random features learning](https://mlsanity.com/paper/2302.00401/) (2023)
- [Bayes-optimal learning of an extensive-width neural network from quadratically many samples](https://mlsanity.com/paper/2408.03733/) (2024)
- [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)
- [On the use of deep learning for phase recovery](https://mlsanity.com/paper/2308.00942/) (2023)
- [Self-Validation: Early Stopping for Single-Instance Deep Generative Priors](https://mlsanity.com/paper/2110.12271/) (2021)
- [Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors](https://mlsanity.com/paper/2106.15358/) (2021)
- [Asymptotics of Learning with Deep Structured (Random) Features](https://mlsanity.com/paper/2402.13999/) (2024)
- [Generative Principal Component Analysis](https://mlsanity.com/paper/2203.09693/) (2022)
- [TurbuGAN: An Adversarial Learning Approach to Spatially-Varying Multiframe Blind Deconvolution with Applications to Imaging Through Turbulence](https://mlsanity.com/paper/2203.06764/) (2022)
- [Score-Guided Intermediate Layer Optimization: Fast Langevin Mixing for Inverse Problems](https://mlsanity.com/paper/2206.09104/) (2022)
- [Bayesian Inversion for Nonlinear Imaging Models using Deep Generative Priors](https://mlsanity.com/paper/2203.10078/) (2022)
- [Non-Iterative Recovery from Nonlinear Observations using Generative Models](https://mlsanity.com/paper/2205.15749/) (2022)
- [Adapt and Diffuse: Sample-adaptive Reconstruction via Latent Diffusion Models](https://mlsanity.com/paper/2309.06642/) (2023)
- [A coherence parameter characterizing generative compressed sensing with Fourier measurements](https://mlsanity.com/paper/2207.09340/) (2022)
- [Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors](https://mlsanity.com/paper/2108.03570/) (2021)
- [Solving Inverse Problems with NerfGANs](https://mlsanity.com/paper/2112.09061/) (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)
- [Non-Iterative Phase Retrieval With Cascaded Neural Networks](https://mlsanity.com/paper/2106.10195/) (2021)
- [Optimization for Amortized Inverse Problems](https://mlsanity.com/paper/2210.13983/) (2022)
- [Inverse problem regularization with hierarchical variational autoencoders](https://mlsanity.com/paper/2303.11217/) (2023)
- [Leveraging Variational Autoencoders for Parameterized MMSE Estimation](https://mlsanity.com/paper/2307.05352/) (2023)
- [Projected Gradient Descent Algorithms for Solving Nonlinear Inverse Problems with Generative Priors](https://mlsanity.com/paper/2209.10093/) (2022)

## Abstract (arXiv)

The phase retrieval problem asks to recover a natural signal y0 ∈ ℝn from m quadratic observations, where m is to be minimized. As is common in many imaging problems, natural signals are considered sparse with respect to a known basis, and the generic sparsity prior is enforced via ℓ1 regularization. While successful in the realm of linear inverse problems, such ℓ1 methods have encountered possibly fundamental limitations, as no computationally efficient algorithm for phase retrieval of a k-sparse signal has been proven to succeed with fewer than O(k2log n) generic measurements, exceeding the theoretical optimum of O(k log n). In this paper, we propose a novel framework for phase retrieval by 1) modeling natural signals as being in the range of a deep generative neural network G : ℝk rightarrow ℝn and 2) enforcing this prior directly by optimizing an empirical risk objective over the domain of the generator. Our formulation has provably favorable global geometry for gradient methods, as soon as m = O(kd2log n), where d is the depth of the network. Specifically, when suitable deterministic conditions on the generator and measurement matrix are met, we construct a descent direction for any point outside of a small neighborhood around the unique global minimizer and its negative multiple, and show that such conditions hold with high probability under Gaussian ensembles of multilayer fully-connected generator networks and measurement matrices. This formulation for structured phase retrieval thus has two advantages over sparsity based methods: 1) deep generative priors can more tightly represent natural signals and 2) information theoretically optimal sample complexity. We corroborate these results with experiments showing that exploiting generative models in phase retrieval tasks outperforms sparse phase retrieval methods.