Electronic Journal of Statistics

A flexible convex relaxation for phase retrieval

Sohail Bahmani and Justin Romberg

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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

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

Received: June 2017
First available in Project Euclid: 15 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F10: Point estimation 90C25: Convex programming
Secondary: 63P30

Phase retrieval Vapnik-Chervonenkis theory convex relaxation

Creative Commons Attribution 4.0 International License.


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

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