# PhaseMax: Convex Phase Retrieval via Basis Pursuit

arXiv: https://arxiv.org/abs/1610.07531 · HTML: https://mlsanity.com/paper/1610.07531/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Oct 2016.

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- [Alternating Phase Langevin Sampling with Implicit Denoiser Priors for Phase Retrieval](https://mlsanity.com/paper/2211.00884/) (2022)
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- [Global convergence of gradient descent for phase retrieval](https://mlsanity.com/paper/2410.09990/) (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)
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## Abstract (arXiv)

We consider the recovery of a (real- or complex-valued) signal from magnitude-only measurements, known as phase retrieval. We formulate phase retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the phase retrieval problem to a higher dimension, PhaseMax is a "non-lifting" relaxation that operates in the original signal dimension. We show that the dual problem to PhaseMax is Basis Pursuit, which implies that phase retrieval can be performed using algorithms initially designed for sparse signal recovery. We develop sharp lower bounds on the success probability of PhaseMax for a broad range of random measurement ensembles, and we analyze the impact of measurement noise on the solution accuracy. We use numerical results to demonstrate the accuracy of our recovery guarantees, and we showcase the efficacy and limits of PhaseMax in practice.