# Phase retrieval

I’ve got the power. / Like the crack of the whip/ I snap attack/ Front to back

January 17, 2017 — November 8, 2019

feature construction

Hilbert space

machine learning

model selection

neural nets

optimization

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 Søndergaard 2016).

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

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## 1 Griffin-Lim iteration

A classic.

## 2 References

Adali, Schreier, and Scharf. 2011. “Complex-Valued Signal Processing: The Proper Way to Deal With Impropriety.”

*IEEE Transactions on Signal Processing*.
Bahmani, and Romberg. 2016. “Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation.”

*arXiv:1610.04210 [Cs, Math, Stat]*.
Barkhuijsen, de Beer, Bovée, et al. 1985. “Retrieval of Frequencies, Amplitudes, Damping Factors, and Phases from Time-Domain Signals Using a Linear Least-Squares Procedure.”

*Journal of Magnetic Resonance (1969)*.
Candes, Emmanuel, Li, and Soltanolkotabi. 2015. “Phase Retrieval via Wirtinger Flow: Theory and Algorithms.”

*IEEE Transactions on Information Theory*.
Candes, Emmanuel J., Strohmer, and Voroninski. 2011. “PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming.”

*Communications on Pure and Applied Mathematics*.
Chen, and Candès. 2015. “Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems.” In

*Advances in Neural Information Processing Systems 28*.
Chi, Lu, and Chen. 2019. “Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview.”

*IEEE Transactions on Signal Processing*.
Dhifallah, Thrampoulidis, and Lu. 2018. “Phase Retrieval via Polytope Optimization: Geometry, Phase Transitions, and New Algorithms.”

*arXiv:1805.09555 [Cs, Math]*.
Goldstein, and Studer. 2016. “PhaseMax: Convex Phase Retrieval via Basis Pursuit.”

*arXiv:1610.07531 [Cs, Math]*.
Griffin, and Lim. 1984. “Signal Estimation from Modified Short-Time Fourier Transform.”

*IEEE Transactions on Acoustics, Speech, and Signal Processing*.
Grosse, Raina, Kwong, et al. 2007. “Shift-Invariant Sparse Coding for Audio Classification.” In

*The Twenty-Third Conference on Uncertainty in Artificial Intelligence (UAI2007)*.
Hand, and Voroninski. 2016. “An Elementary Proof of Convex Phase Retrieval in the Natural Parameter Space via the Linear Program PhaseMax.”

*arXiv:1611.03935 [Cs, Math]*.
Jaganathan, Eldar, and Hassibi. 2015. “Phase Retrieval: An Overview of Recent Developments.”

*arXiv:1510.07713 [Cs, Math]*.
Jagatap, and Hegde. 2018. “Towards Sample-Optimal Methods for Solving Random Quadratic Equations with Structure.” In

*2018 IEEE International Symposium on Information Theory (ISIT)*.
Li, and Zhou. 2016. “On Gradient Descent Algorithm for Generalized Phase Retrieval Problem.”

*arXiv:1607.01121 [Math]*.
Luo, Alghamdi, and Lu. 2019. “Optimal Spectral Initialization for Signal Recovery with Applications to Phase Retrieval.”

*IEEE Transactions on Signal Processing*.
Marafioti, Perraudin, Holighaus, et al. 2019. “Adversarial Generation of Time-Frequency Features with Application in Audio Synthesis.” In

*International Conference on Machine Learning*.
Masuyama, Yatabe, Koizumi, et al. 2019. “Deep Griffin-Lim Iteration.” In

*ICASSP 2019*.
Moravec, Romberg, and Baraniuk. 2007. “Compressive Phase Retrieval.” In

*Wavelets XII*.
Perraudin, Balazs, and Sondergaard. 2013. “A Fast Griffin-Lim Algorithm.” In

*2013 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics*.
Pinilla, Bacca, Tourneret, et al. 2018. “A Smoothing Stochastic Phase Retrieval Algorithm for Solving Random Quadratic Systems.” In

*2018 IEEE Statistical Signal Processing Workshop (SSP)*.
Pru, and Søndergaard. 2016. “Real-Time Spectrogram Inversion Using Phase Gradient Heap Integration.”

Rajaei, Gigan, Krzakala, et al. 2017. “Robust Phase Retrieval with the Swept Approximate Message Passing (prSAMP) Algorithm.”

*Image Processing On Line*.
Seuret, and Gouaisbaut. 2013. “Wirtinger-Based Integral Inequality: Application to Time-Delay Systems.”

*Automatica*.
Shechtman, Beck, and Eldar. 2014. “GESPAR: Efficient Phase Retrieval of Sparse Signals.”

*IEEE Transactions on Signal Processing*.
Shechtman, Eldar, Cohen, et al. 2015. “Phase Retrieval with Application to Optical Imaging: A Contemporary Overview.”

*IEEE Signal Processing Magazine*.
Slaney, Naar, and Lyon. 1994. “Auditory Model Inversion for Sound Separation.” In

*Proceedings of ICASSP ’94.*
Sturm, Daudet, and Roads. 2006. “Pitch-Shifting Audio Signals Using Sparse Atomic Approximations.” In

*Proceedings of the 1st ACM Workshop on Audio and Music Computing Multimedia*. AMCMM ’06.
Wang, Giannakis, Saad, et al. 2017. “Solving Most Systems of Random Quadratic Equations.” In

*Advances in Neural Information Processing Systems 30*.
Zhang, Chi, and Liang. 2016. “Provable Non-Convex Phase Retrieval with Outliers: Median Truncated Wirtinger Flow.”

*arXiv:1603.03805 [Stat]*.
Zhang, and Liang. 2016. “Reshaped Wirtinger Flow for Solving Quadratic System of Equations.” In

*Advances in Neural Information Processing Systems 29*.