# Acceleration and Implicit Regularization in Gaussian Phase Retrieval

arXiv: https://arxiv.org/abs/2311.12888 · HTML: https://mlsanity.com/paper/2311.12888/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 62 references that are in the map.
- First posted on arXiv 21 Nov 2023.

## Builds on (most cited first, 29 of 29)

- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Implicit Regularization for Optimal Sparse Recovery](https://mlsanity.com/paper/1909.05122/) (2019)
- [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)
- [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)
- [High-Dimensional Linear Regression via Implicit Regularization](https://mlsanity.com/paper/1903.09367/) (2019)
- [Implicit regularization in Heavy-ball momentum accelerated stochastic gradient descent](https://mlsanity.com/paper/2302.00849/) (2023)
- [Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval](https://mlsanity.com/paper/1705.02356/) (2017)
- [Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming](https://mlsanity.com/paper/1310.0807/) (2013)
- [Implicit Regularization of Accelerated Methods in Hilbert Spaces](https://mlsanity.com/paper/1905.13000/) (2019)
- [Nonconvex Matrix Factorization from Rank-One Measurements](https://mlsanity.com/paper/1802.06286/) (2018)
- [Recursive Importance Sketching for Rank Constrained Least Squares: Algorithms and High-order Convergence](https://mlsanity.com/paper/2011.08360/) (2020)
- [Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensing](https://mlsanity.com/paper/2204.11516/) (2022)
- [Practical Phase Retrieval Using Double Deep Image Priors](https://mlsanity.com/paper/2211.00799/) (2022)
- [A Continuous-Time Mirror Descent Approach to Sparse Phase Retrieval](https://mlsanity.com/paper/2010.10168/) (2020)
- [Fast Global Convergence for Low-rank Matrix Recovery via Riemannian Gradient Descent with Random Initialization](https://mlsanity.com/paper/2012.15467/) (2020)
- [Sparse Phase Retrieval via Truncated Amplitude Flow](https://mlsanity.com/paper/1611.07641/) (2016)
- [The local convexity of solving systems of quadratic equations](https://mlsanity.com/paper/1506.07868/) (2015)
- [Hadamard Wirtinger Flow for Sparse Phase Retrieval](https://mlsanity.com/paper/2006.01065/) (2020)
- [Accelerated Wirtinger Flow: A fast algorithm for ptychography](https://mlsanity.com/paper/1806.05546/) (2018)
- [Quickly Finding a Benign Region via Heavy Ball Momentum in Non-Convex Optimization](https://mlsanity.com/paper/2010.01449/) (2020)
- [Bures-Wasserstein Barycenters and Low-Rank Matrix Recovery](https://mlsanity.com/paper/2210.14671/) (2022)
- [Analytical Convergence Regions of Accelerated Gradient Descent in Nonconvex Optimization under Regularity Condition](https://mlsanity.com/paper/1810.03229/) (2018)
- [Holographic Phase Retrieval and Reference Design](https://mlsanity.com/paper/1901.06453/) (2019)
- [Towards Low-Photon Nanoscale Imaging: Holographic Phase Retrieval via Maximum Likelihood Optimization](https://mlsanity.com/paper/2105.11512/) (2021)

## Built on by (most cited first, 1 of 1)

- [A Smoothing Newton Method for Rank-one Matrix Recovery](https://mlsanity.com/paper/2507.23017/) (2025)

## Abstract (arXiv)

We study accelerated optimization methods in the Gaussian phase retrieval problem. In this setting, we prove that gradient methods with Polyak or Nesterov momentum have similar implicit regularization to gradient descent. This implicit regularization ensures that the algorithms remain in a nice region, where the cost function is strongly convex and smooth despite being nonconvex in general. This ensures that these accelerated methods achieve faster rates of convergence than gradient descent. Experimental evidence demonstrates that the accelerated methods converge faster than gradient descent in practice.