# Phase Retrieval, Spectral Factorization, and Applications

Recover phaseless complex signals from measured magnitudes, or recover complex signals from the auto-correlation sequences. While for 1D signals the problem is solved both in theory and in practice, that for 2D or higher-D’s are still largely open. Emphasis will be put on recent developments. (**Update: Mar 14 2021**)

[**S**] indicates my contribution.

### Provable Results with Algorithms

- Stochasticity helps to navigate rough landscapes: comparing gradient-descent-based algorithms in the phase retrieval problem (2021)
- Optimal Sample Complexity of Gradient Descent for Amplitude Flow via Non-Lipschitz Matrix Concentration (2020)
- Compressive Phase Retrieval: Optimal Sample Complexity with Deep Generative Priors (2020)
- Phase retrieval of complex-valued objects via a randomized Kaczmarz method (2020)
- A Deterministic Convergence Framework for Exact Non-Convex Phase Retrieval (2020)
- The recovery of complex sparse signals from few phaseless measurements (2019)
- Conjugate Phase Retrieval in Paley-Wiener Space (2019)
- Online Stochastic Gradient Descent with Arbitrary Initialization Solves Non-smooth, Non-convex Phase Retrieval (2019)
- On phase retrieval via matrix completion and the estimation of low rank PSD matrices (2019)
- Complex phase retrieval from subgaussian measurements (2019)
- Solving Random Systems of Quadratic Equations with Tanh Wirtinger Flow (2019)
- On the Global Minimizers of Real Robust Phase Retrieval with Sparse Noise (2019)
- Solving a perturbed amplitude-based model for phase retrieval (2019)
- Spectral Method for Phase Retrieval: an Expectation Propagation Perspective (2019)
- Rigorous Analysis of Spectral Methods for Random Orthogonal Matrices (2019)
- Solving Complex Quadratic Systems with Full-Rank Random Matrices (2019)
- Dual-Reference Design for Holographic Coherent Diffraction Imaging ([
**S**], 2019) - Holographic Phase Retrieval and Optimal Reference Design ([
**S**], 2019) - A Generalization of Wirtinger Flow for Exact Interferometric Inversion (2019)
- Phase Retrieval by Alternating Minimization with Random Initialization (2018)
- Optimal Spectral Initialization for Signal Recovery with Applications to Phase Retrieval (2018)
- A Proximal Operator for Multispectral Phase Retrieval Problems (2018)
- Towards the optimal construction of a loss function without spurious local minima for solving quadratic equations (2018)
- Solving systems of phaseless equations via Riemannian optimization with optimal sampling complexity (2018)
- Linear Spectral Estimators and an Application to Phase Retrieval (2018)
- Approximate Message Passing for Amplitude Based Optimization (2018)
- Solving Systems of Quadratic Equations via Exponential-type Gradient Descent Algorithm (2018)
- Phase Retrieval via Polytope Optimization: Geometry, Phase Transitions, and New Algorithms (2018)
- Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval (2018)
- PhaseLin: Linear Phase Retrieval (2018)
- A Precise Analysis of PhaseMax in Phase Retrieval (2018)
- Optimization-based AMP for Phase Retrieval: The Impact of Initialization and $\ell_2$-regularization (2018)
- Compressive Phase Retrieval of Structured Signal (2017)
- Misspecified Nonconvex Statistical Optimization for Phase Retrieval (2017)
- Compressive Phase Retrieval via Reweighted Amplitude Flow (2017)
- A Local Analysis of Block Coordinate Descent for Gaussian Phase Retrieval ([
**S**], 2017) - Convolutional Phase Retrieval via Gradient Descent (2017)
- Linear Convergence of An Iterative Phase Retrieval Algorithm with Data Reuse (2017)
- The nonsmooth landscape of phase retrieval (2017)
- Phase Retrieval via Linear Programming: Fundamental Limits and Algorithmic Improvements (2017)
- Convergence of the randomized Kaczmarz method for phase retrieval (2017)
- Phase Retrieval via Randomized Kaczmarz: Theoretical Guarantees (2017)
- Phase retrieval using alternating minimization in a batch setting (2017)
- Solving Almost all Systems of Random Quadratic Equations (2017)
- Phase Retrieval Using Structured Sparsity: A Sample Efficient Algorithmic Framework (2017)
- Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval (2017)
- Robust Wirtinger Flow for Phase Retrieval with Arbitrary Corruption (2017)
- Phase Retrieval via Sparse Wirtinger Flow (2017)
- Structured signal recovery from quadratic measurements: Breaking sample complexity barriers via nonconvex optimization (2017)
- Multiple Illumination Phaseless Super-Resolution (MIPS) with Applications To Phaseless DOA Estimation and Diffraction Imaging (2017)
- Sparse phase retrieval of one-dimensional signals by Prony’s method (2017)
- Corruption Robust Phase Retrieval via Linear Programming (2016)
- Phase Retrieval from Local Measurements: Improved Robustness via Eigenvector-Based Angular Synchronization (2016)
- Sparse Phase Retrieval via Truncated Amplitude Flow (2016)
- Compressed Sensing from Phaseless Gaussian Measurements via Linear Programming in the Natural Parameter Spac (2016)
- An Elementary Proof of Convex Phase Retrieval in the Natural Parameter Space via the Linear Program PhaseMax (2016)
- Solving Large-scale Systems of Random Quadratic Equations via Stochastic Truncated Amplitude Flow (2016)
- Low rank matrix recovery from Clifford orbits (2016)
- PhaseMax: Convex Phase Retrieval via Basis Pursuit (2016)
- Phase Retrieval From the Magnitudes of Affine Linear Measurements (2016)
- Phaseless super-resolution in the continuous domain (2016)
- Reconstruction of signals from their autocorrelation and cross-correlation vectors, with applications to phase retrieval and blind channel estimation (2016)
- Low Rank Phase Retrieval (2016)
- Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation (2016)
- Phase retrieval with random Gaussian sensing vectors by alternating projections (2016)
- Non-Convex Phase Retrieval from STFT Measurements (2016)
- Gauss-Newton Method for Phase Retrieval (2016)
- Phase Retrieval via Incremental Truncated Wirtinger Flow (2016)
- Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow (2016)
- Reshaped Wirtinger Flow for Solving Quadratic Systems of Equations (2016)
- Provable Non-convex Phase Retrieval with Outliers: Median Truncated Wirtinger Flow (2016)
- Phase Retrieval from 1D Fourier Measurements: Convexity, Uniqueness, and Algorithms (2016)
- A Geometric Analysis of Phase Retrieval ([
**S**], 2016) - Phaseless super-resolution using masks (2016)
- A Least Squares Approach for Stable Phase Retrieval from Short-Time Fourier Transform Magnitude (2015)
- STFT Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms (2015)
- The Local Convexity of Solving Quadratic Equations (2015)
- Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems (2015)
- Solving systems of phaseless equations via Kaczmarz methods: A proof of concept study (2015)
- Phase Retrieval via Wirtinger Flow: Theory and Algorithms (2014)
- Phase Retrieval using Alternating Minimization (2013)
- Phase Retrieval from Coded Diffraction Patterns (2013)
- Solving Quadratic Equations via PhaseLift when There Are About As Many Equations As Unknowns (2012)
- PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming (2011)
- Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms (2010)

### Recovery Limits

- Stable Phase Retrieval from Locally Stable and Conditionally Connected Measurements (2020)
- Phase retrieval in high dimensions: Statistical and computational phase transitions (2020)
- Toward a mathematical theory of the crystallographic phase retrieval problem (2020)
- Information Theoretic Limits for Phase Retrieval with Subsampled Haar Sensing Matrices (2019)
- Phase Retrieval for Binary Signals: Box Relaxation and Uniqueness (2019)
- Support Recovery in the Phase Retrieval Model: Information-Theoretic Fundamental Limits (2019)
- Geometry of the Phase Retrieval Problem (2018)
- Blind Phaseless Short-Time Fourier Transform Recovery (2018)
- Coded Aperture Ptychography: Uniqueness and Reconstruction (2017)
- Fundamental Limits of Weak Recovery with Applications to Phase Retrieval (2017)
- Fundamental Limits of PhaseMax for Phase Retrieval: A Replica Analysis (2017)
- On Signal Reconstruction from FROG Measurements (2017)
- On the Uniqueness of FROG Methods (2017)
- On The 2D Phase Retrieval Problem (2016)
- Generalized phase retrieval : measurement number, matrix recovery and beyond (2016)
- Ambiguities in One-Dimensional Discrete Phase Retrieval from Fourier Magnitudes (2015)
- Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case (2015)
- Phase retrieval for the Cauchy wavelet transform (2014)
- Phase Transitions in Phase Retrieval (2014)
- Phase retrieval from very few measurements (2013)
- Invertibility and Robustness of Phaseless Reconstruction (2013)
- Phase Retrieval: Stability and Recovery Guarantees (2012)
- Stability of unique Fourier-transform phase reconstruction (1983)
- Reducible polynomials in more than one variable (1982)
- The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform (1982)

### Experimental/Partial Analytic Results

- Inverse Problems, Deep Learning, and Symmetry Breaking ([
**S**], 2020) - Phase Retrieval for Partially Coherent Observations (2020)
- Convex combination of alternating projection and Douglas-Rachford operators for phase retrieval (2020)
- Fixed Point Analysis of Douglas-Rachford Splitting for Ptychography and Phase Retrieval (2019)
- A note on Douglas-Rachford, subgradients, and phase retrieval (2019)
- Learning to Synthesize: Robust Phase Retrieval at Low Photon counts (2019)
- Don’t take it lightly: Phasing optical random projections with unknown operators (2019)
- Frequency-Resolved Optical Gating Recovery via Smoothing Gradient (2019)
- Optimization on Spheres: Models and Proximal Algorithms with Computational Performance Comparisons (2018)
- Phase retrieval for Bragg coherent diffraction imaging at high X-ray energies (2018)
- Spectral Method for Multiplexed Phase Retrieval and Application in Optical Imaging in Complex Media (2018)
- FASPR: A fast sparse phase retrieval algorithm via the epigraph concept (2018)
- Constrained phase retrieval: when alternating projection meets regularization (2018)
- Blind Ptychographic Phase Retrieval via Convergent Alternating Direction Method of Multipliers (2018)
- Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval from Minimal Measurements (2018)
- Phase Retrieval Under a Generative Prior (2018)
- Generalized Proximal Smoothing for Phase Retrieval (2018)
- PhasePack User Guide (2017)
- Non-iterative Filter Bank Phase (Re)Construction (2017)
- Toward High-Quality Real-Time Signal Reconstruction From STFT Magnitude (2017)
- A Noniterative Method for Reconstruction of Phase From STFT Magnitude (2017)
- Coordinate Descent Algorithms for Phase Retrieval (2017)
- Benchmark problems for phase retrieval (2017)
- A general framework for denoising phaseless diffraction measurements (2016)
- The Phase Retrieval Toolbox (2016)
- Real-time spectrogram inversion using phase gradient heap integration (2016)
- Phase retrieval for wavelet transforms (2015)
- Fourier Phase Retrieval with a Single Mask by Douglas-Rachford Algorithm (2015)
- Experimental robustness of Fourier Ptychographic phase retrieval algorithms (2015)
- Phase retrieval for wavelet transforms (2015)
- Sparse Phase Retrieval from Short-Time Fourier Measurements (2014)
- Phase retrieval for imaging problems (2013)
- Phase Recovery, MaxCut and Complex Semidefinite Programming (2012)
- Alternating direction methods for classical and ptychographic phase retrieval (2012)
- Phase Retrieval via Matrix Completion (2011)
- Phase retrieval and saddle-point optimization (2006)
- A unified evaluation of iterative projection algorithms for phase retrieval (2006)
- Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization (2002)

### Survey/Introduction Articles

- The Numerics of Phase Retrieval (2020)
- Fourier Phase Retrieval: Uniqueness and Algorithms (2017)
- Phase Retrieval: An Overview of Recent Developments (2015)
- Phase Retrieval with Application to Optical Imaging (2014)
- The phase retrieval problem (1995)

Disclaimer- This page is meant to serve a hub for references on this problem, and does not represent in any way personal endorsement of papers listed here. So I do not hold any responsibility for quality and technical correctness of each paper listed here. The reader is advised to use this resource with discretion.

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