mlsanityData to 7 Oct 2026

Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow

arXiv 1506.03382 ↗First posted 10 Jun 2015math.ST cs.IT cs.NAcited by 23

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Abstract (arXiv)

This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal x ∈ ℝp from noisy quadratic measurements yj = (aj' x )2 + εj, j=1, ldots, m, with independent sub-exponential noise εj. The goals are to understand the effect of the sparsity of x on the estimation precision and to construct a computationally feasible estimator to achieve the optimal rates. Inspired by the Wirtinger Flow [12] proposed for noiseless and non-sparse phase retrieval, 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 aj's are independent standard Gaussian random vectors, provided that the sample size is sufficiently large compared to the sparsity of x.

Key facts. Cited by 23 papers in the map (data through 7 Oct 2026). First posted on arXiv 10 Jun 2015.

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Citing papers per year: 2021: 5; 2022: 3; 2023: 6; 2024: 3; 2025: 5; 2026: 1.

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  1. 2022Convergence of gradient descent for deep neural networkscited by 24
  2. 2021Optimal Gradient-based Algorithms for Non-concave Bandit Optimizationcited by 15
  3. 2021Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priorscited by 14
  4. 2021Bandit Phase Retrievalcited by 12
  5. 2022Sudakov-Fernique post-AMP, and a new proof of the local convexity of the TAP free energycited by 9
  6. 2026Manifold constrained steepest descent for smooth and closed-set optimizationcited by 8
  7. 2022Misspecified Phase Retrieval with Generative Priorscited by 7
  8. 2023Statistical-Computational Tradeoffs in Mixed Sparse Linear Regressioncited by 7
  9. 2023Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priorscited by 4
  10. 2023Near Optimal Heteroscedastic Regression with Symbiotic Learningcited by 3
  11. 2025Optimal sparse phase retrieval via a quasi-Bayesian approachcited by 3
  12. 2021Total Least Squares Phase Retrievalcited by 2
11 more
  1. 2021Robust High-Dimensional Regression with Coefficient Thresholding and its Application to Imaging Data Analysiscited by 1
  2. 2023Poisson-Gaussian Holographic Phase Retrieval with Score-based Image Priorcited by 1
  3. 2023Recovering Simultaneously Structured Data via Non-Convex Iteratively Reweighted Least Squarescited by 1
  4. 2023Mode-wise Principal Subspace Pursuit and Matrix Spiked Covariance Modelcited by 1
  5. 2024Computationally efficient reductions between some statistical modelscited by 0
  6. 2024Low Complexity Regularized Phase Retrievalcited by 0
  7. 2024Smoothed Robust Phase Retrievalcited by 0
  8. 2025AltGDmin: Alternating GD and Minimization for Partly-Decoupled (Federated) Optimizationcited by 0
  9. 2025Phase transition of descending phase retrieval algorithmscited by 0
  10. 2025Optimal spectral initializers impact on phase retrieval phase transitions -- an RDT viewcited by 0
  11. 2025Phase retrieval with rank d measurements -- descending algorithms phase transitionscited by 0