# Noisy phase retrieval from subgaussian measurements

arXiv: https://arxiv.org/abs/2412.07401 · HTML: https://mlsanity.com/paper/2412.07401/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2024.

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

- [NS-RGS: Newton-Schulz based Riemannian gradient method for orthogonal group synchronization](https://mlsanity.com/paper/2604.07372/) (2026)
- [Statistical Inference on Gradient Flows](https://mlsanity.com/paper/2606.01257/) (2026)

## Abstract (arXiv)

This paper aims to address the phase retrieval problem from subgaussian measurements with arbitrary noise, with a focus on devising robust and efficient algorithms for solving non-convex problems. To ensure uniqueness of solutions in the subgaussian setting, we explore two commonly used assumptions: either the subgaussian measurements satisfy a fourth-moment condition or the target signals exhibit non-peakiness. For each scenario, we introduce a novel spectral initialization method that yields robust initial estimates. Building on this, we employ leave-one-out arguments to show that the classical Wirtinger flow algorithm achieves a linear rate of convergence for both real-valued and complex-valued cases, provided the sampling complexity m≥ O(n log3 m), where n is the dimension of the underlying signals. In contrast to existing work, our algorithms are regularization-free, requiring no truncation, trimming, or additional penalty terms, and they permit the algorithm step sizes as large as O(1), compared to the O(1/n) in previous literature. Furthermore, our results accommodate arbitrary noise vectors that meet certain statistical conditions, covering a wide range of noise scenarios, with sub-exponential noise as a notable special case. The effectiveness of our algorithms is validated through various numerical experiments. We emphasize that our findings provide the first theoretical guarantees for recovering non-peaky signals using non-convex methods from Bernoulli measurements, which is of independent interest.