# Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation

arXiv: https://arxiv.org/abs/1610.04210 · HTML: https://mlsanity.com/paper/1610.04210/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Oct 2016.

## Built on by (most cited first, 11 of 11)

- [Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors](https://mlsanity.com/paper/2106.15358/) (2021)
- [Misspecified Phase Retrieval with Generative Priors](https://mlsanity.com/paper/2210.05571/) (2022)
- [Stochastic Light Field Holography](https://mlsanity.com/paper/2307.06277/) (2023)
- [Max-Linear Regression by Convex Programming](https://mlsanity.com/paper/2103.07020/) (2021)
- [Interacting Particle Systems on Networks: joint inference of the network and the interaction kernel](https://mlsanity.com/paper/2402.08412/) (2024)
- [A Sample Efficient Alternating Minimization-based Algorithm For Robust Phase Retrieval](https://mlsanity.com/paper/2409.04733/) (2024)
- [Learning Multi-type heterogeneous interacting particle systems](https://mlsanity.com/paper/2602.03954/) (2026)
- [Towards Designing Optimal Sensing Matrices for Generalized Linear Inverse Problems](https://mlsanity.com/paper/2111.03237/) (2021)
- [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 propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with "lifting" and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techniques for phase retrieval. We relax the quadratic equations for phaseless measurements to inequality constraints each of which representing a symmetric "slab". Through a simple convex program, our proposed estimator finds an extreme point of the intersection of these slabs that is best aligned with a given anchor vector. We characterize geometric conditions that certify success of the proposed estimator. Furthermore, using classic results in statistical learning theory, we show that for random measurements the geometric certificates hold with high probability at an optimal sample complexity. Phase transition of our estimator is evaluated through simulations. Our numerical experiments also suggest that the proposed method can solve phase retrieval problems with coded diffraction measurements as well.