# Solving phase retrieval with random initial guess is nearly as good as by spectral initialization

arXiv: https://arxiv.org/abs/2101.03540 · HTML: https://mlsanity.com/paper/2101.03540/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2021.

## Built on by (most cited first, 9 of 9)

- [Stochasticity helps to navigate rough landscapes: comparing gradient-descent-based algorithms in the phase retrieval problem](https://mlsanity.com/paper/2103.04902/) (2021)
- [Bayes-optimal learning of an extensive-width neural network from quadratically many samples](https://mlsanity.com/paper/2408.03733/) (2024)
- [Misspecified Phase Retrieval with Generative Priors](https://mlsanity.com/paper/2210.05571/) (2022)
- [The Role of the Time-Dependent Hessian in High-Dimensional Optimization](https://mlsanity.com/paper/2403.02418/) (2024)
- [Overparametrization bends the landscape: BBP transitions at initialization in simple Neural Networks](https://mlsanity.com/paper/2510.18435/) (2025)
- [The Local Landscape of Phase Retrieval Under Limited Samples](https://mlsanity.com/paper/2311.15221/) (2023)
- [Topological Exploration of High-Dimensional Empirical Risk Landscapes: general approach, and applications to phase retrieval](https://mlsanity.com/paper/2602.17779/) (2026)
- [Smoothed Robust Phase Retrieval](https://mlsanity.com/paper/2409.01570/) (2024)
- [Global convergence of gradient descent for phase retrieval](https://mlsanity.com/paper/2410.09990/) (2024)

## Abstract (arXiv)

The problem of recovering a signal x∈ ℝn from a set of magnitude measurements yi=|langle ai, x rangle |, ; i=1,ldots,m is referred as phase retrieval, which has many applications in fields of physical sciences and engineering. In this paper we show that the smoothed amplitude flow model for phase retrieval has benign geometric structure under the optimal sampling complexity. In particular, we show that when the measurements ai∈ ℝn are Gaussian random vectors and the number of measurements m≥ Cn, our smoothed amplitude flow model has no spurious local minimizers with high probability, ie., the target solution x is the unique global minimizer (up to a global phase) and the loss function has a negative directional curvature around each saddle point. Due to this benign geometric landscape, the phase retrieval problem can be solved by the gradient descent algorithms without spectral initialization. Numerical experiments show that the gradient descent algorithm with random initialization performs well even comparing with state-of-the-art algorithms with spectral initialization in empirical success rate and convergence speed.