PhaseMax swMATH ID: 24954 Software Authors: Goldstein, Tom; Studer, Christoph Description: PhaseMax: convex phase retrieval via basis pursuit. We consider the recovery of a (real- or complex-valued) signal from magnitude-only measurements, known as phase retrieval. We formulate phase retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the phase retrieval problem to a higher dimension, PhaseMax is a ”non-lifting” relaxation that operates in the original signal dimension. We show that the dual problem to PhaseMax is Basis Pursuit, which implies that phase retrieval can be performed using algorithms initially designed for sparse signal recovery. We develop sharp lower bounds on the success probability of PhaseMax for a broad range of random measurement ensembles, and we analyze the impact of measurement noise on the solution accuracy. We use numerical results to demonstrate the accuracy of our recovery guarantees, and we showcase the efficacy and limits of PhaseMax in practice. Homepage: https://arxiv.org/abs/1610.07531 Related Software: Wirtinger Flow; PhaseLift; GESPAR; BranchHull; SparsePR; PDCO; PhasePack; PhaseCode; CoSaMP; LIBSVM; RRR; BlockPR; Macaulay2; ProxToolbox; Superflip; iPiano; FASTA; FMS; ADMiRA; minFunc Cited in: 30 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year PhaseMax: convex phase retrieval via basis pursuit. Zbl 1390.94194Goldstein, Tom; Studer, Christoph 2018 all top 5 Cited by 66 Authors 3 Bahmani, Sohail 3 Hand, Paul E. 3 Li, Huiping 3 Li, Song 2 Aghasi, Alireza 2 Cai, Jian-Feng 2 Liu, Haixia 2 Wang, Yang 1 Alacaoglu, Ahmet 1 Bendory, Tamir 1 Bonifaci, Vincenzo 1 Cevher, Volkan 1 Chang, Huibin 1 Charisopoulos, Vasileios 1 Chen, Yudong 1 Chen, Yuxin 1 Chi, Yuejie 1 Davis, Damek 1 Díaz, Mateo 1 Ding, Lijun 1 Drusvyatskiy, Dmitriy 1 Duan, Yuping 1 Edidin, Dan 1 Fercoq, Olivier 1 Gao, Bing 1 Goldstein, Tom 1 Gui, Guan 1 Joshi, Babhru 1 Krahmer, Felix 1 Lavaei, Javad 1 Lee, Kiryung 1 Li, Jingzhi 1 Lin, Shijian 1 Lou, Yifei 1 Lu, Xiliang 1 Luke, David Russell 1 Luo, Qi 1 Ma, Cong 1 Marchesini, Stefano 1 Mondelli, Marco 1 Montanari, Andrea 1 Qu, Qing 1 Romberg, Justin K. 1 Ruan, Guoqing 1 Sabach, Shoham 1 Sojoudi, Somayeh 1 Stöger, Dominik 1 Studer, Christoph 1 Sun, Ju 1 Teboulle, Marc 1 Voroninski, Vladislav 1 Wang, Hongxia 1 Wang, Kaixin 1 Wang, Kaizheng 1 Wang, Ya 1 Xia, Yu 1 Xiang, Jianhong 1 Xiao, Zhuolei 1 Yazici, Birsen 1 Yin, Xiangjun 1 Yonel, Bariscan 1 You, Juntao 1 Yue, Huihui 1 Zhang, Richard Y. 1 Zhang, Teng 1 Zhuang, Zhitao all top 5 Cited in 20 Serials 5 Foundations of Computational Mathematics 4 Applied and Computational Harmonic Analysis 2 Journal of Machine Learning Research (JMLR) 2 SIAM Journal on Imaging Sciences 2 SIAM Journal on Mathematics of Data Science 1 IEEE Transactions on Information Theory 1 Journal of the Franklin Institute 1 SIAM Journal on Matrix Analysis and Applications 1 Journal of Scientific Computing 1 SIAM Journal on Optimization 1 Computational Optimization and Applications 1 Applied Mathematics. Series B (English Edition) 1 Advances in Computational Mathematics 1 The Journal of Fourier Analysis and Applications 1 Communications in Mathematical Sciences 1 Analysis and Applications (Singapore) 1 International Journal of Wavelets, Multiresolution and Information Processing 1 Inverse Problems in Science and Engineering 1 Electronic Journal of Statistics 1 Symmetry all top 5 Cited in 14 Fields 21 Operations research, mathematical programming (90-XX) 19 Information and communication theory, circuits (94-XX) 8 Numerical analysis (65-XX) 7 Statistics (62-XX) 4 Harmonic analysis on Euclidean spaces (42-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 4 Optics, electromagnetic theory (78-XX) 2 Probability theory and stochastic processes (60-XX) 1 Commutative algebra (13-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Approximations and expansions (41-XX) 1 Functional analysis (46-XX) 1 Differential geometry (53-XX) 1 Computer science (68-XX) Citations by Year