## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 11, Number 2 (2017), 5254-5281.

### A flexible convex relaxation for phase retrieval

Sohail Bahmani and Justin Romberg

#### Abstract

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. We demonstrate the effectiveness of the proposed method through numerical simulations using both independent random measurements and coded diffraction patterns.

We also extend this formulation to include *sparsity constraints* on the target vector. With this additional constraint, we show that, considering “nested” measurements, the number of phaseless measurements needed to recover the sparse vector is essentially the same (to within a logarithmic factor) as the number of linear measurements needed by standard compressive sensing techniques.

#### Article information

**Source**

Electron. J. Statist., Volume 11, Number 2 (2017), 5254-5281.

**Dates**

Received: June 2017

First available in Project Euclid: 15 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1513306873

**Digital Object Identifier**

doi:10.1214/17-EJS1378SI

**Mathematical Reviews number (MathSciNet)**

MR3738211

**Zentralblatt MATH identifier**

06825046

**Subjects**

Primary: 62F10: Point estimation 90C25: Convex programming

Secondary: 63P30

**Keywords**

Phase retrieval Vapnik-Chervonenkis theory convex relaxation

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Bahmani, Sohail; Romberg, Justin. A flexible convex relaxation for phase retrieval. Electron. J. Statist. 11 (2017), no. 2, 5254--5281. doi:10.1214/17-EJS1378SI. https://projecteuclid.org/euclid.ejs/1513306873