You know the power of the signal; what is the phase? Griffin-Lim algorithm, Wirtinger flow methods based on Wirtinger calculus, Phase-gradient heap integration (Pru and SΓΈndergaard 2016).

TODO: investigate Yue M Luβs work on phase retrieval as an important example in a large classe of somewhat- analytically-understood nonconvex problems, starting from his recent slide deck on that theme.

π

## References

Adali, T., P. J. Schreier, and L. L. Scharf. 2011. βComplex-Valued Signal Processing: The Proper Way to Deal With Impropriety.β

*IEEE Transactions on Signal Processing*59 (11): 5101β25.Bahmani, Sohail, and Justin Romberg. 2016. βPhase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation.β

*arXiv:1610.04210 [Cs, Math, Stat]*, October.Barkhuijsen, H., R. de Beer, W. M. J. BovΓ©e, and D. van Ormondt. 1985. βRetrieval of Frequencies, Amplitudes, Damping Factors, and Phases from Time-Domain Signals Using a Linear Least-Squares Procedure.β

*Journal of Magnetic Resonance (1969)*61 (3): 465β81.Candes, Emmanuel J., Thomas Strohmer, and Vladislav Voroninski. 2011. βPhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming.β

*Communications on Pure and Applied Mathematics*66 (8): 1241β74.Candes, Emmanuel, Xiaodong Li, and Mahdi Soltanolkotabi. 2015. βPhase Retrieval via Wirtinger Flow: Theory and Algorithms.β

*IEEE Transactions on Information Theory*61 (4): 1985β2007.Chen, Yuxin, and Emmanuel CandΓ¨s. 2015. βSolving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems.β In

*Advances in Neural Information Processing Systems 28*, edited by C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett, 739β47. Curran Associates, Inc.Chi, Yuejie, Yue M. Lu, and Yuxin Chen. 2019. βNonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview.β

*IEEE Transactions on Signal Processing*67 (20): 5239β69.Dhifallah, Oussama, Christos Thrampoulidis, and Yue M. Lu. 2018. βPhase Retrieval via Polytope Optimization: Geometry, Phase Transitions, and New Algorithms.β

*arXiv:1805.09555 [Cs, Math]*, May.Goldstein, Tom, and Christoph Studer. 2016. βPhaseMax: Convex Phase Retrieval via Basis Pursuit.β

*arXiv:1610.07531 [Cs, Math]*, October.Griffin, D., and Jae Lim. 1984. βSignal Estimation from Modified Short-Time Fourier Transform.β

*IEEE Transactions on Acoustics, Speech, and Signal Processing*32 (2): 236β43.Grosse, Roger, Rajat Raina, Helen Kwong, and Andrew Y. Ng. 2007. βShift-Invariant Sparse Coding for Audio Classification.β In

*The Twenty-Third Conference on Uncertainty in Artificial Intelligence (Uai2007)*, 9:8.Hand, Paul, and Vladislav Voroninski. 2016. βAn Elementary Proof of Convex Phase Retrieval in the Natural Parameter Space via the Linear Program PhaseMax.β

*arXiv:1611.03935 [Cs, Math]*, November.Jaganathan, Kishore, Yonina C. Eldar, and Babak Hassibi. 2015. βPhase Retrieval: An Overview of Recent Developments.β

*arXiv:1510.07713 [Cs, Math]*, October.Jagatap, G., and C. Hegde. 2018. βTowards Sample-Optimal Methods for Solving Random Quadratic Equations with Structure.β In

*2018 IEEE International Symposium on Information Theory (ISIT)*, 1000:2296β2300.Li, Ji, and Tie Zhou. 2016. βOn Gradient Descent Algorithm for Generalized Phase Retrieval Problem.β

*arXiv:1607.01121 [Math]*, July.Luo, Wangyu, Wael Alghamdi, and Yue M. Lu. 2019. βOptimal Spectral Initialization for Signal Recovery with Applications to Phase Retrieval.β

*IEEE Transactions on Signal Processing*67 (9): 2347β56.Marafioti, AndrΓ©s, NathanaΓ«l Perraudin, Nicki Holighaus, and Piotr Majdak. 2019. βAdversarial Generation of Time-Frequency Features with Application in Audio Synthesis.β In

*International Conference on Machine Learning*, 4352β62. PMLR.Masuyama, Yoshiki, Kohei Yatabe, Yuma Koizumi, Yasuhiro Oikawa, and Noboru Harada. 2019. βDeep Griffin-Lim Iteration.β In

*ICASSP 2019*.Moravec, Matthew L., Justin K. Romberg, and Richard G. Baraniuk. 2007. βCompressive Phase Retrieval.β In

*Wavelets XII*, 6701:670120. San Diego, CA: International Society for Optics and Photonics.Perraudin, Nathanael, Peter Balazs, and Peter L. Sondergaard. 2013. βA Fast Griffin-Lim Algorithm.β In

*2013 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics*, 1β4. New Paltz, NY: IEEE.Pinilla, S., J. Bacca, J. Tourneret, and H. Arguello. 2018. βA Smoothing Stochastic Phase Retrieval Algorithm for Solving Random Quadratic Systems.β In

*2018 IEEE Statistical Signal Processing Workshop (SSP)*, 278β82.Pru, ZdeneΛk, and Peter L SΓΈndergaard. 2016. βReal-Time Spectrogram Inversion Using Phase Gradient Heap Integration,β 5.

Rajaei, Boshra, Sylvain Gigan, Florent Krzakala, and Laurent Daudet. 2017. βRobust Phase Retrieval with the Swept Approximate Message Passing (prSAMP) Algorithm.β

*Image Processing On Line*7 (January): 43β55.Seuret, Alexandre, and FrΓ©dΓ©ric Gouaisbaut. 2013. βWirtinger-Based Integral Inequality: Application to Time-Delay Systems.β

*Automatica*49 (9): 2860β66.Shechtman, Y., A. Beck, and Y. C. Eldar. 2014. βGESPAR: Efficient Phase Retrieval of Sparse Signals.β

*IEEE Transactions on Signal Processing*62 (4): 928β38.Shechtman, Y., Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev. 2015. βPhase Retrieval with Application to Optical Imaging: A Contemporary Overview.β

*IEEE Signal Processing Magazine*32 (3): 87β109.Slaney, M., D. Naar, and R.E. Lyon. 1994. βAuditory Model Inversion for Sound Separation.β In

*Proceedings of ICASSP β94.*, ii:II/77β80. Adelaide, SA, Australia: IEEE.Sturm, Bob L., Laurent Daudet, and Curtis Roads. 2006. βPitch-Shifting Audio Signals Using Sparse Atomic Approximations.β In

*Proceedings of the 1st ACM Workshop on Audio and Music Computing Multimedia*, 45β52. AMCMM β06. New York, NY, USA: ACM.Wang, Gang, Georgios Giannakis, Yousef Saad, and Jie Chen. 2017. βSolving Most Systems of Random Quadratic Equations.β In

*Advances in Neural Information Processing Systems 30*, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, 1867β77. Curran Associates, Inc.Zhang, Huishuai, Yuejie Chi, and Yingbin Liang. 2016. βProvable Non-Convex Phase Retrieval with Outliers: Median Truncated Wirtinger Flow.β

*arXiv:1603.03805 [Stat]*, March.Zhang, Huishuai, and Yingbin Liang. 2016. βReshaped Wirtinger Flow for Solving Quadratic System of Equations.β In

*Advances in Neural Information Processing Systems 29*, edited by D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, 2622β30. Curran Associates, Inc.
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