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A Precise Analysis of PhaseMax in Phase Retrieval

Salehi, Fariborz and Abbasi, Ehsan and Hassibi, Babak (2018) A Precise Analysis of PhaseMax in Phase Retrieval. In: 2018 IEEE International Symposium on Information Theory (ISIT). IEEE , Piscataway, NJ, pp. 976-980. ISBN 978-1-5386-4780-6.

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Recovering an unknown complex signal from the magnitude of linear combinations of the signal is referred to as phase retrieval. We present an exact performance analysis of a recently proposed convex-optimization-formulation for this problem, known as PhaseMax. Standard convex-relaxation-based methods in phase retrieval resort to the idea of “lifting” which makes them computationally inefficient, since the number of unknowns is effectively squared. In contrast, PhaseMax is a novel convex relaxation that does not increase the number of unknowns. Instead it relies on an initial estimate of the true signal which must be externally provided. In this paper, we investigate the required number of measurements for exact recovery of the signal in the large system limit and when the linear measurement matrix is random with iid standard normal entries. If n denotes the dimension of the unknown complex signal and m the number of phaseless measurements, then in the large system limit, m/n > 4/cos^2(θ) measurements is necessary and sufficient to recover the signal with high probability, where θ is the angle between the initial estimate and the true signal. Our result indicates a sharp phase transition in the asymptotic regime which matches the empirical result in numerical simulations.

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Additional Information:© 2018 IEEE. This work was inspired by the ideas presented in [20]. The authors would like to thank Yue M. Lu, Christos Thrampoulidis, and Philipp Walk for helpful discussions.
Record Number:CaltechAUTHORS:20181126-143043881
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Official Citation:F. Salehi, E. Abbasi and B. Hassibi, "A Precise Analysis of PhaseMax in Phase Retrieval," 2018 IEEE International Symposium on Information Theory (ISIT), Vail, CO, 2018, pp. 976-980. doi: 10.1109/ISIT.2018.8437494
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:91187
Deposited By: Tony Diaz
Deposited On:26 Nov 2018 22:37
Last Modified:03 Oct 2019 20:32

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