# Provable Phase Retrieval with Mirror Descent

arXiv: https://arxiv.org/abs/2210.09248 · HTML: https://mlsanity.com/paper/2210.09248/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 74 references that are in the map.
- First posted on arXiv 17 Oct 2022.

## Builds on (most cited first, 20 of 20)

- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval](https://mlsanity.com/paper/1803.07726/) (2018)
- [Phase Retrieval using Alternating Minimization](https://mlsanity.com/paper/1306.0160/) (2013)
- [Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems](https://mlsanity.com/paper/1505.05114/) (2015)
- [First Order Methods beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problems](https://mlsanity.com/paper/1706.06461/) (2017)
- [Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow](https://mlsanity.com/paper/1605.08285/) (2016)
- [Phase Retrieval from Coded Diffraction Patterns](https://mlsanity.com/paper/1310.3240/) (2013)
- [The Numerics of Phase Retrieval](https://mlsanity.com/paper/2004.05788/) (2020)
- [The nonsmooth landscape of phase retrieval](https://mlsanity.com/paper/1711.03247/) (2017)
- [Phase retrieval with random Gaussian sensing vectors by alternating projections](https://mlsanity.com/paper/1609.03088/) (2016)
- [Convex-Concave Backtracking for Inertial Bregman Proximal Gradient Algorithms in Non-Convex Optimization](https://mlsanity.com/paper/1904.03537/) (2019)
- [PhasePack: A Phase Retrieval Library](https://mlsanity.com/paper/1711.10175/) (2017)
- [A Stochastic Bregman Primal-Dual Splitting Algorithm for Composite Optimization](https://mlsanity.com/paper/2112.11928/) (2021)
- [Transversality and alternating projections for nonconvex sets](https://mlsanity.com/paper/1401.7569/) (2014)
- [On local convergence of the method of alternating projections](https://mlsanity.com/paper/1312.5681/) (2013)
- [Quantitative convergence analysis of iterated expansive, set-valued mappings](https://mlsanity.com/paper/1605.05725/) (2016)
- [Linear Convergence of the Douglas-Rachford Method for Two Closed Sets](https://mlsanity.com/paper/1401.6509/) (2014)
- [Convergence Analysis of the Relaxed Douglas-Rachford Algorithm](https://mlsanity.com/paper/1811.11590/) (2018)

## Built on by (most cited first, 2 of 2)

- [Stable Phase Retrieval with Mirror Descent](https://mlsanity.com/paper/2405.10754/) (2024)
- [Low Complexity Regularized Phase Retrieval](https://mlsanity.com/paper/2407.16413/) (2024)

## Abstract (arXiv)

In this paper, we consider the problem of phase retrieval, which consists of recovering an n-dimensional real vector from the magnitude of its m linear measurements. We propose a mirror descent (or Bregman gradient descent) algorithm based on a wisely chosen Bregman divergence, hence allowing to remove the classical global Lipschitz continuity requirement on the gradient of the non-convex phase retrieval objective to be minimized. We apply the mirror descent for two random measurements: the iid standard Gaussian and those obtained by multiple structured illuminations through Coded Diffraction Patterns (CDP). For the Gaussian case, we show that when the number of measurements m is large enough, then with high probability, for almost all initializers, the algorithm recovers the original vector up to a global sign change. For both measurements, the mirror descent exhibits a local linear convergence behaviour with a dimension-independent convergence rate. Our theoretical results are finally illustrated with various numerical experiments, including an application to the reconstruction of images in precision optics.