# 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: Nov 06 2020**)

[**S**] indicates my contribution.

### Provable Results with Algorithms

- 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.

If you’d like your paper to be listed here- Just drop me a few lines via email (which can be found on “Welcome” page). If you don’t bother to spend a word, just deposit your paper on arXiv. I get email alert about new animals there every morning, and will be happy to hunt one for this zoo if it seemsfit.