# Phase retrieval with random Gaussian sensing vectors by alternating projections

arXiv: https://arxiv.org/abs/1609.03088 · HTML: https://mlsanity.com/paper/1609.03088/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2016.

## Built on by (most cited first, 15 of 15)

- [Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstruction](https://mlsanity.com/paper/2106.15013/) (2021)
- [Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensing](https://mlsanity.com/paper/2204.11516/) (2022)
- [Sharp global convergence guarantees for iterative nonconvex optimization: A Gaussian process perspective](https://mlsanity.com/paper/2109.09859/) (2021)
- [Asymptotic Dynamics of Alternating Minimization for Bilinear Regression](https://mlsanity.com/paper/2402.04751/) (2024)
- [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)
- [Poisson-Gaussian Holographic Phase Retrieval with Score-based Image Prior](https://mlsanity.com/paper/2305.07712/) (2023)
- [Agnostic Learning of Mixed Linear Regressions with EM and AM Algorithms](https://mlsanity.com/paper/2406.01149/) (2024)
- [A Dual Basis Approach for Structured Robust Euclidean Distance Geometry](https://mlsanity.com/paper/2505.18414/) (2025)
- [Tractable Gaussian Phase Retrieval with Heavy Tails and Adversarial Corruption with Near-Linear Sample Complexity](https://mlsanity.com/paper/2601.18245/) (2026)
- [Global convergence of gradient descent for phase retrieval](https://mlsanity.com/paper/2410.09990/) (2024)
- [Stability properties of gradient flow dynamics for the symmetric low-rank matrix factorization problem](https://mlsanity.com/paper/2411.15972/) (2024)
- [Phase transition of descending phase retrieval algorithms](https://mlsanity.com/paper/2506.18275/) (2025)
- [Optimal spectral initializers impact on phase retrieval phase transitions -- an RDT view](https://mlsanity.com/paper/2506.18279/) (2025)
- [Phase retrieval with rank d measurements -- descending algorithms phase transitions](https://mlsanity.com/paper/2506.18282/) (2025)

## Abstract (arXiv)

We consider a phase retrieval problem, where we want to reconstruct a n-dimensional vector from its phaseless scalar products with m sensing vectors. We assume the sensing vectors to be independently sampled from complex normal distributions. We propose to solve this problem with the classical non-convex method of alternating projections. We show that, when m≥ Cn for C large enough, alternating projections succeed with high probability, provided that they are carefully initialized. We also show that there is a regime in which the stagnation points of the alternating projections method disappear, and the initialization procedure becomes useless. However, in this regime, m has to be of the order of n2. Finally, we conjecture from our numerical experiments that, in the regime m=O(n), there are stagnation points, but the size of their attraction basin is small if m/n is large enough, so alternating projections can succeed with probability close to 1 even with no special initialization.