On Lipschitz analysis and Lipschitz synthesis for the phase retrieval problem

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© 2016 Elsevier Inc. All rights reserved. We prove two results with regard to reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem"). First we show that phase retrievable nonlinear maps are bi-Lipschitz with respect to appropriate metrics on the quotient space. Specifically, if nonlinear analysis maps α,β:H→→ℝm are injective, with α(x)=(|<x,fk>|)km=1 and β(x)=(|<x,fk>|2)km=1, where {f1,...,fm} is a frame for a Hilbert space H and H=H/T1, then α is bi-Lipschitz with respect to the class of "natural metrics" Dp(x,y)=minφ||x-eiφy||p, whereas β is bi-Lipschitz with respect to the class of matrix-norm induced metrics dp(x,y)=||xx∗-yy∗||p. Second we prove that reconstruction can be performed using Lipschitz continuous maps. That is, there exist left inverse maps (synthesis maps) ω,ψ:ℝm→H of α and β respectively, that are Lipschitz continuous with respect to appropriate metrics. Additionally, we obtain the Lipschitz constants of ω and ψ in terms of the lower Lipschitz constants of α and β, respectively. Surprisingly, the increase in both Lipschitz constants is a relatively small factor, independent of the space dimension or the frame redundancy.





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Balan, R, and D Zou (2016). On Lipschitz analysis and Lipschitz synthesis for the phase retrieval problem. Linear Algebra and Its Applications, 496. pp. 152–181. 10.1016/j.laa.2015.12.029 Retrieved from https://hdl.handle.net/10161/21934.

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Dongmian Zou

Assistant Professor of Data Science at Duke Kunshan University

Dongmian Zou received the B.S. degree in Mathematics (First Honour) from the Chinese University of Hong Kong in 2012 and the Ph.D. degree in Applied Mathematics and Scientific Computation from the University of Maryland, College Park in 2017. From 2017 to 2020, he served as a post-doctorate researcher at the Institute for Mathematics and its Applications, and the School of Mathematics at the University of Minnesota, Twin Cities. He joined Duke Kunshan University in 2020 where he is currently an Assistant Professor of Data Science in the Division of Natural and Applied Sciences. He is also affiliated with the the Zu Chongzhi Center for Mathematics and Computational Sciences (CMCS) and the Data Science Research Center (DSRC). His research is in the intersection of applied harmonic analysis, machine learning and signal processing. His current interest includes geometric deep learning, robustness, anomaly detection, and applications in e.g., communication, circuits and medical imaging.

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