# Robust Gradient Descent for Phase Retrieval

arXiv: https://arxiv.org/abs/2410.10623 · HTML: https://mlsanity.com/paper/2410.10623/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 41 references that are in the map.
- First posted on arXiv 14 Oct 2024.

## Builds on (most cited first, 25 of 25)

- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Agnostic Estimation of Mean and Covariance](https://mlsanity.com/paper/1604.06968/) (2016)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Sever: A Robust Meta-Algorithm for Stochastic Optimization](https://mlsanity.com/paper/1803.02815/) (2018)
- [Robust Estimation via Robust Gradient Estimation](https://mlsanity.com/paper/1802.06485/) (2018)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution](https://mlsanity.com/paper/1711.10467/) (2017)
- [Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems](https://mlsanity.com/paper/1505.05114/) (2015)
- [Outlier Robust Mean Estimation with Subgaussian Rates via Stability](https://mlsanity.com/paper/2007.15618/) (2020)
- [Robust regression with covariate filtering: Heavy tails and adversarial contamination](https://mlsanity.com/paper/2009.12976/) (2020)
- [Sub-Gaussian estimators of the mean of a random vector](https://mlsanity.com/paper/1702.00482/) (2017)
- [Robust and Heavy-Tailed Mean Estimation Made Simple, via Regret Minimization](https://mlsanity.com/paper/2007.15839/) (2020)
- [Mean Estimation with Sub-Gaussian Rates in Polynomial Time](https://mlsanity.com/paper/1809.07425/) (2018)
- [Defending Against Saddle Point Attack in Byzantine-Robust Distributed Learning](https://mlsanity.com/paper/1806.05358/) (2018)
- [Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval](https://mlsanity.com/paper/1705.02356/) (2017)
- [The Numerics of Phase Retrieval](https://mlsanity.com/paper/2004.05788/) (2020)
- [Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tails](https://mlsanity.com/paper/2205.08494/) (2022)
- [Bandit Phase Retrieval](https://mlsanity.com/paper/2106.01660/) (2021)
- [Efficient learning with robust gradient descent](https://mlsanity.com/paper/1706.00182/) (2017)
- [Nearly Minimax-Optimal Rates for Noisy Sparse Phase Retrieval via Early-Stopped Mirror Descent](https://mlsanity.com/paper/2105.03678/) (2021)
- [Robust Methods for High-Dimensional Linear Learning](https://mlsanity.com/paper/2208.05447/) (2022)
- [Median-Truncated Nonconvex Approach for Phase Retrieval with Outliers](https://mlsanity.com/paper/1603.03805/) (2016)
- [Robust supervised learning with coordinate gradient descent](https://mlsanity.com/paper/2201.13372/) (2022)
- [Error Bound of Empirical ℓ2 Risk Minimization for Noisy Standard and Generalized Phase Retrieval Problems](https://mlsanity.com/paper/2205.13827/) (2022)

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

- [Tractable Gaussian Phase Retrieval with Heavy Tails and Adversarial Corruption with Near-Linear Sample Complexity](https://mlsanity.com/paper/2601.18245/) (2026)
- [Convex Basins in Single-Index Model Loss Landscapes: Applications to Robust Recovery under Strong Adversarial Corruption](https://mlsanity.com/paper/2605.29497/) (2026)

## Abstract (arXiv)

Recent progress in robust statistical learning has mainly tackled convex problems, like mean estimation or linear regression, with non-convex challenges receiving less attention. Phase retrieval exemplifies such a non-convex problem, requiring the recovery of a signal from only the magnitudes of its linear measurements, without phase (sign) information. While several non-convex methods, especially those involving the Wirtinger Flow algorithm, have been proposed for noiseless or mild noise settings, developing solutions for heavy-tailed noise and adversarial corruption remains an open challenge. In this paper, we investigate an approach that leverages robust gradient descent techniques to improve the Wirtinger Flow algorithm's ability to simultaneously cope with fourth moment bounded noise and adversarial contamination in both the inputs (covariates) and outputs (responses). We address two scenarios: known zero-mean noise and completely unknown noise. For the latter, we propose a preprocessing step that alters the problem into a new format that does not fit traditional phase retrieval approaches but can still be resolved with a tailored version of the algorithm for the zero-mean noise context.