Solving systems of random quadratic equations via truncated amplitude flow

Gang Wang, Georgios B. Giannakis, Yonina C. Eldar

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88 Scopus citations


This paper presents a new algorithm, termed truncated amplitude flow (TAF), to recover an unknown vector x from a system of quadratic equations of the form yi=|{ai, x}|2, where a-i's are given random measurement vectors. This problem is known to be NP-hard in general. We prove that as soon as the number of equations is on the order of the number of unknowns, TAF recovers the solution exactly (up to a global unimodular constant) with high probability and complexity growing linearly with both the number of unknowns and the number of equations. Our TAF approach adapts the amplitude-based empirical loss function and proceeds in two stages. In the first stage, we introduce an orthogonality-promoting initialization that can be obtained with a few power iterations. Stage two refines the initial estimate by successive updates of scalable truncated generalized gradient iterations, which are able to handle the rather challenging nonconvex and nonsmooth amplitude-based objective function. In particular, when vectors x and ai's are real valued, our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version. Numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations. Furthermore, even under the same initialization, the proposed amplitude-based refinement outperforms existing Wirtinger flow variants, corroborating the superior performance of TAF over state-of-the-art algorithms.

Original languageEnglish (US)
Article number8049465
Pages (from-to)773-794
Number of pages22
JournalIEEE Transactions on Information Theory
Issue number2
StatePublished - Feb 2018

Bibliographical note

Funding Information:
Manuscript received July 3, 2016; revised April 1, 2017; accepted July 25, 2017. Date of publication September 26, 2017; date of current version January 18, 2018. G. Wang and G. B. Giannakis were supported by NSF under Grant 1500713 and Grant 1514056. This paper was presented at the 2016 Neural Information Processing Systems Conference.

Funding Information:
His research interests focus on the areas of high-dimensional statistical signal processing, stochastic and nonconvex optimization with applications to autonomous energy grids, and deep learning. He received a National Scholarship from China in 2014, the Student Travel Awards from the NSF (2016) and NIPS (2017), and a Best Student Paper Award at the 2017 European Signal Processing Conference.

Funding Information:
His general interests span the areas of communications, networking and statistical signal processing - subjects on which he has published more than 400 journal papers, 700 conference papers, 25 book chapters, two edited books and two research monographs (h-index 127). Current research focuses on learning from Big Data, wireless cognitive radios, and network science with applications to social, brain, and power networks with renewables. He is the (co-) inventor of 30 patents issued, and the (co-) recipient of 8 best paper awards from the IEEE Signal Processing (SP) and Communications Societies, including the G. Marconi Prize Paper Award in Wireless Communications. He also received Technical Achievement Awards from the SP Society (2000), from EURASIP (2005), a Young Faculty Teaching Award, the G. W. Taylor Award for Distinguished Research from the University of Minnesota, and the IEEE Fourier Technical Field Award (2015). He is a Fellow of EURASIP, and has served the IEEE in a number of posts, including that of a Distinguished Lecturer for the IEEE-SP Society.

Funding Information:
Dr. Eldar has received many awards for excellence in research and teaching, including the IEEE Signal Processing Society Technical Achievement Award (2013), the IEEE/AESS Fred Nathanson Memorial Radar Award (2014), and the IEEE Kiyo Tomiyasu Award (2016). She was a Horev Fellow of the Leaders in Science and Technology program at the Technion and an Alon Fellow. She received the Michael Bruno Memorial Award from the Rothschild Foundation, the Weizmann Prize for Exact Sciences, the Wolf Foundation Krill Prize for Excellence in Scientific Research, the Henry Taub Prize for Excellence in Research (twice), the Hershel Rich Innovation Award (three times), the Award for Women with Distinguished Contributions, the Andre and Bella Meyer Lectureship, the Career Development Chair at the Technion, the Muriel & David Jacknow Award for Excellence in Teaching, and the Technions Award for Excellence in Teaching (two times). She received several best paper awards and best demo awards together with her research students and colleagues including the SIAM outstanding Paper Prize, the UFFC Outstanding Paper Award, the Signal Processing Society Best Paper Award and the IET Circuits, Devices and Systems Premium Award, and was selected as one of the 50 most influential women in Israel.


  • Amplitude-based cost function
  • Linear convergence to global minimum
  • Nonconvex optimization
  • Orthogonality-promoting initialization
  • Phase retrieval
  • Truncated gradient

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