Phase Retrieval via Wirtinger Flow: Theory and Algorithms. We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal x of C^n about which we have phaseless samples of the form y_r = |< a_r,x >|^2, r = 1,2,...,m (knowledge of the phase of these samples would yield a linear system). This paper develops a non-convex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a near-linear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of non-convex optimization schemes that may have implications for computational problems beyond phase retrieval.
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References in zbMATH (referenced in 110 articles , 1 standard article )
Showing results 101 to 110 of 110.
- Chang, Huibin; Lou, Yifei; Ng, Michael K.; Zeng, Tieyong: Phase retrieval from incomplete magnitude information via total variation regularization (2016)
- Friedlander, Michael P.; Macêdo, Ives: Low-rank spectral optimization via gauge duality (2016)
- Iwen, Mark A.; Viswanathan, Aditya; Wang, Yang: Fast phase retrieval from local correlation measurements (2016)
- Kabanava, Maryia; Kueng, Richard; Rauhut, Holger; Terstiege, Ulrich: Stable low-rank matrix recovery via null space properties (2016)
- Peng, Wei; Wang, Hongxia: Binary sparse phase retrieval via simulated annealing (2016)
- Wei, Ke; Cai, Jian-Feng; Chan, Tony F.; Leung, Shingyu: Guarantees of Riemannian optimization for low rank matrix recovery (2016)
- Candès, Emmanuel J.; Li, Xiaodong; Soltanolkotabi, Mahdi: Phase retrieval via Wirtinger flow: theory and algorithms (2015)
- Führ, Hartmut; Rzeszotnik, Ziemowit: On biunimodular vectors for unitary matrices (2015)
- Ling, Shuyang; Strohmer, Thomas: Self-calibration and biconvex compressive sensing (2015)
- Wei, Ke: Solving systems of phaseless equations via Kaczmarz methods: a proof of concept study (2015)