# The Numerics of Phase Retrieval

arXiv: https://arxiv.org/abs/2004.05788 · HTML: https://mlsanity.com/paper/2004.05788/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Apr 2020.

## Built on by (most cited first, 21 of 21)

- [Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem](https://mlsanity.com/paper/2101.08286/) (2021)
- [Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2112.04330/) (2021)
- [From Stability to Chaos: Analyzing Gradient Descent Dynamics in Quadratic Regression](https://mlsanity.com/paper/2310.01687/) (2023)
- [Practical Phase Retrieval Using Double Deep Image Priors](https://mlsanity.com/paper/2211.00799/) (2022)
- [Image Reconstruction without Explicit Priors](https://mlsanity.com/paper/2303.12217/) (2023)
- [Discovering Structure From Corruption for Unsupervised Image Reconstruction](https://mlsanity.com/paper/2304.05589/) (2023)
- [Spectral Estimators for Structured Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2308.14507/) (2023)
- [Stable Phase Retrieval with Mirror Descent](https://mlsanity.com/paper/2405.10754/) (2024)
- [Provable Phase Retrieval with Mirror Descent](https://mlsanity.com/paper/2210.09248/) (2022)
- [Fast and Robust Phase Retrieval via Deep Expectation-Consistent Approximation](https://mlsanity.com/paper/2407.09687/) (2024)
- [Robust Gradient Descent for Phase Retrieval](https://mlsanity.com/paper/2410.10623/) (2024)
- [DDRM-PR: Fourier Phase Retrieval using Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2501.03030/) (2025)
- [WARPd: A linearly convergent first-order method for inverse problems with approximate sharpness conditions](https://mlsanity.com/paper/2110.12437/) (2021)
- [Integrating Generative and Physics-Based Models for Ptychographic Imaging with Uncertainty Quantification](https://mlsanity.com/paper/2412.10882/) (2024)
- [Position-Blind Ptychography: Viability of image reconstruction via data-driven variational inference](https://mlsanity.com/paper/2509.25269/) (2025)
- [Machine learning for classifying and interpreting coherent X-ray speckle patterns](https://mlsanity.com/paper/2211.08194/) (2022)
- [Low Complexity Regularized Phase Retrieval](https://mlsanity.com/paper/2407.16413/) (2024)
- [Sharper Convergence Rates for Nonconvex Optimisation via Reduction Mappings](https://mlsanity.com/paper/2506.08428/) (2025)
- [prNet: Data-Driven Phase Retrieval via Stochastic Refinement](https://mlsanity.com/paper/2507.09608/) (2025)
- [I2I-PR: Deep Iterative Refinement for Phase Retrieval using Image-to-Image Diffusion Models](https://mlsanity.com/paper/2507.09609/) (2025)
- [Deeply unrolled alternating projections for phase retrieval](https://mlsanity.com/paper/2610.17316/) (2026)

## Abstract (arXiv)

Phase retrieval, i.e., the problem of recovering a function from the squared magnitude of its Fourier transform, arises in many applications such as X-ray crystallography, diffraction imaging, optics, quantum mechanics, and astronomy. This problem has confounded engineers, physicists, and mathematicians for many decades. Recently, phase retrieval has seen a resurgence in research activity, ignited by new imaging modalities and novel mathematical concepts. As our scientific experiments produce larger and larger datasets and we aim for faster and faster throughput, it becomes increasingly important to study the involved numerical algorithms in a systematic and principled manner. Indeed, the last decade has witnessed a surge in the systematic study of computational algorithms for phase retrieval. In this paper we will review these recent advances from a numerical viewpoint.