mlsanityData to 7 Oct 2026

Phase Retrieval via Wirtinger Flow: Theory and Algorithms

arXiv 1407.1065 ↗First posted 3 Jul 2014cs.IT cs.NA math.FAcited by 123

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Abstract (arXiv)

We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal x of C^n about which we have phaseless samples of the form y_r = |< a_r,x >|^2, r = 1,2,...,m (knowledge of the phase of these samples would yield a linear system). This paper develops a non-convex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a near-linear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of non-convex optimization schemes that may have implications for computational problems beyond phase retrieval.

Key facts. Cited by 123 papers in the map (data through 7 Oct 2026). First posted on arXiv 3 Jul 2014.

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Citing papers per year: 2021: 22; 2022: 27; 2023: 20; 2024: 20; 2025: 22; 2026: 12.

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  1. 2021Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstructioncited by 77
  2. 2024Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstructioncited by 73
  3. 2023Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Modelscited by 60
  4. 2021Large Learning Rate Tames Homogeneity: Convergence and Balancing Effectcited by 48
  5. 2021Hessian Eigenspectra of More Realistic Nonlinear Modelscited by 33
  6. 2021Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurementscited by 28
  7. 2023On the use of deep learning for phase recoverycited by 20
  8. 2021Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passingcited by 18
  9. 2022Group-invariant max filteringcited by 18
  10. 2025Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approachcited by 18
  11. 2022Restarted Nonconvex Accelerated Gradient Descent: No More Polylogarithmic Factor in the O(ε-7/4) Complexitycited by 17
  12. 2021Optimal Gradient-based Algorithms for Non-concave Bandit Optimizationcited by 15
38 more
  1. 2021Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priorscited by 14
  2. 2023Escaping mediocrity: how two-layer networks learn hard generalized linear models with SGDcited by 14
  3. 2022Improved Global Guarantees for the Nonconvex Burer--Monteiro Factorization via Rank Overparameterizationcited by 13
  4. 2021Bandit Phase Retrievalcited by 12
  5. 2022Generative Principal Component Analysiscited by 12
  6. 2025Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorizationcited by 12
  7. 2022Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensingcited by 11
  8. 2021Sharp global convergence guarantees for iterative nonconvex optimization: A Gaussian process perspectivecited by 10
  9. 2022Bayesian Inversion for Nonlinear Imaging Models using Deep Generative Priorscited by 9
  10. 2022Non-Iterative Recovery from Nonlinear Observations using Generative Modelscited by 9
  11. 2022Sudakov-Fernique post-AMP, and a new proof of the local convexity of the TAP free energycited by 9
  12. 2023PtychoDV: Vision Transformer-Based Deep Unrolling Network for Ptychographic Image Reconstructioncited by 9
  13. 2023Catapult Dynamics and Phase Transitions in Quadratic Netscited by 8
  14. 2024Guaranteed Nonconvex Factorization Approach for Tensor Train Recoverycited by 8
  15. 2022SiSPRNet: End-to-End Learning for Single-Shot Phase Retrievalcited by 7
  16. 2022Misspecified Phase Retrieval with Generative Priorscited by 7
  17. 2023Statistical-Computational Tradeoffs in Mixed Sparse Linear Regressioncited by 7
  18. 2022Accelerating SGD for Highly Ill-Conditioned Huge-Scale Online Matrix Completioncited by 6
  19. 2023Provably Accelerating Ill-Conditioned Low-rank Estimation via Scaled Gradient Descent, Even with Overparameterizationcited by 6
  20. 2025Provable Benefits of Unsupervised Pre-training and Transfer Learning via Single-Index Modelscited by 6
  21. 2021Nearly Minimax-Optimal Rates for Noisy Sparse Phase Retrieval via Early-Stopped Mirror Descentcited by 5
  22. 2022Matrix Completion with Hierarchical Graph Side Informationcited by 5
  23. 2022From Gradient Flow on Population Loss to Learning with Stochastic Gradient Descentcited by 5
  24. 2023Fast and Accurate Estimation of Low-Rank Matrices from Noisy Measurements via Preconditioned Non-Convex Gradient Descentcited by 5
  25. 2023Spectral Estimators for Structured Generalized Linear Models via Approximate Message Passingcited by 5
  26. 2026Spectral Gradient Descent Mitigates Anisotropy-Driven Misalignment: A Case Study in Phase Retrievalcited by 5
  27. 2021Householder Dice: A Matrix-Free Algorithm for Simulating Dynamics on Gaussian and Random Orthogonal Ensemblescited by 4
  28. 2021Dynamic Proximal Unrolling Network for Compressive Imagingcited by 4
  29. 2021STORM+: Fully Adaptive SGD with Momentum for Nonconvex Optimizationcited by 4
  30. 2023Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priorscited by 4
  31. 2024Computational and Statistical Guarantees for Tensor-on-Tensor Regression with Tensor Train Decompositioncited by 4
  32. 2024Sample and Computationally Efficient Robust Learning of Gaussian Single-Index Modelscited by 4
  33. 2025A Scalable Factorization Approach for High-Order Structured Tensor Recoverycited by 4
  34. 2025PnP-CM: Consistency Models as Plug-and-Play Priors for Inverse Problemscited by 4
  35. 2021Phase Retrieval using Expectation Consistent Signal Recovery Algorithm based on Hypernetworkcited by 3
  36. 2021Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimizationcited by 3
  37. 2022Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspectivecited by 3
  38. 2022Fundamental Limits of Two-layer Autoencoders, and Achieving Them with Gradient Methodscited by 3

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