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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