A Geometric Analysis of Phase Retrieval

  title={A Geometric Analysis of Phase Retrieval},
  author={Ju Sun and Qing Qu and John Wright},
  journal={Foundations of Computational Mathematics},
Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, $$y_k = \left| \varvec{a}_k^* \varvec{x} \right| $$yk=ak∗x for $$k = 1, \ldots , m$$k=1,…,m, is it possible to recover $$\varvec{x} \in \mathbb C^n$$x∈Cn (i.e., length-n complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well… 
Construction of optimal spectral methods in phase retrieval
This paper combines the linearization of message-passing algorithms and the analysis of the Bethe Hessian, a classical tool of statistical physics, to derive optimal spectral methods for arbitrary channel noise and right-unitarily invariant matrix $\mathbf{\Phi}$, in an automated manner.
Solving phase retrieval with random initial guess is nearly as good as by spectral initialization
Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval
Extensive numerical tests show that the proposed convex relaxation semidefinite programming (SDP) approach outperforms other state-of-the-art methods in the sharpest phase transition of perfect recovery for Gaussian model and the best reconstruction quality for other non-Gaussian models, in particular Fourier phase retrieval.
Phase retrieval in high dimensions: Statistical and computational phase transitions
This work derives sharp asymptotics for the lowest possible estimation error achievable statistically and unveil the existence of sharp phase transitions for the weak- and full-recovery thresholds as a function of the singular values of the matrix $\mathbf{\Phi}$.
Complex phase retrieval from subgaussian measurements
This paper proves that even when a subgaussian vector $ x_0 \in \mathbb{C}^n $ does not fulfill a small-ball probability assumption, the PhaseLift method is still able to reconstruct a large class of signals from the measurements.
Phase Retrieval Using Alternating Minimization in a Batch Setting
  • Teng Zhang
  • Computer Science
    2018 Information Theory and Applications Workshop (ITA)
  • 2018
A modified alternating minimization method in a batch setting is proposed, and it is proved that when ${m}=O({n}{log^{3}n})$, the proposed algorithm with random initialization recovers the underlying signal with high probability.
Convolutional Phase Retrieval
This model is motivated by applications to channel estimation, optics, and underwater acoustic communication, where the signal of interest is acted on by a given channel/filter, and phase information is difficult or impossible to acquire.
Sparse Signal Recovery From Phaseless Measurements via Hard Thresholding Pursuit
Gradient descent with random initialization: fast global convergence for nonconvex phase retrieval
This paper provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval, without the need of (i) carefully-designed initialization, (ii) sample splitting, or (iii) sophisticated saddle-point escaping schemes.
Performance bound of the intensity-based model for noisy phase retrieval
These results are the first theoretical guarantees for the intensity-based model and its sparse version and under the assumption of $m \gtrsim d$ and $\mathbf{a}_j, j=1,\ldots,m,$ being Gaussian random vectors.


Phase recovery, MaxCut and complex semidefinite programming
This work casts the phase retrieval problem as a non-convex quadratic program over a complex phase vector and formulates a tractable relaxation similar to the classical MaxCut semidefinite program.
Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow
A novel thresholded gradient descent algorithm is proposed and it is shown to adaptively achieve the minimax optimal rates of convergence over a wide range of sparsity levels when the a_j's are independent standard Gaussian random vectors, provided that the sample size is sufficiently large compared to the sparsity of $x.
PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
It is shown that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques, and it is proved that the methodology is robust vis‐à‐vis additive noise.
GESPAR: Efficient Phase Retrieval of Sparse Signals
This work proposes a fast local search method for recovering a sparse signal from measurements of its Fourier transform (or other linear transform) magnitude which it refers to as GESPAR: GrEedy Sparse PhAse Retrieval, which does not require matrix lifting, unlike previous approaches, and therefore is potentially suitable for large scale problems such as images.
More Algorithms for Provable Dictionary Learning
The current paper designs algorithms that allow sparsity up to $n/poly(\log n)$.
Phase Retrieval via Wirtinger Flow: Theory and Algorithms
This paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm that is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements.
New Algorithms for Learning Incoherent and Overcomplete Dictionaries
This paper presents a polynomial-time algorithm for learning overcomplete dictionaries; the only previously known algorithm with provable guarantees is the recent work of Spielman, Wang and Wright who gave an algorithm for the full-rank case.
Phase Retrieval Using Alternating Minimization
This work represents the first theoretical guarantee for alternating minimization (albeit with resampling) for any variant of phase retrieval problems in the non-convex setting.
Non-Convex Phase Retrieval From STFT Measurements
It is shown that in some cases the unique solution can be obtained by the principal eigenvector of a matrix, constructed as the solution of a simple least-squares problem, and it is proved that under appropriate conditions, the proposed initialization is close to the underlying signal.