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

arXiv: https://arxiv.org/abs/1506.03382 · HTML: https://mlsanity.com/paper/1506.03382/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2015.

## Built on by (most cited first, 23 of 23)

- [Convergence of gradient descent for deep neural networks](https://mlsanity.com/paper/2203.16462/) (2022)
- [Optimal Gradient-based Algorithms for Non-concave Bandit Optimization](https://mlsanity.com/paper/2107.04518/) (2021)
- [Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors](https://mlsanity.com/paper/2106.15358/) (2021)
- [Bandit Phase Retrieval](https://mlsanity.com/paper/2106.01660/) (2021)
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- [Manifold constrained steepest descent for smooth and closed-set optimization](https://mlsanity.com/paper/2601.21487/) (2026)
- [Misspecified Phase Retrieval with Generative Priors](https://mlsanity.com/paper/2210.05571/) (2022)
- [Statistical-Computational Tradeoffs in Mixed Sparse Linear Regression](https://mlsanity.com/paper/2303.02118/) (2023)
- [Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priors](https://mlsanity.com/paper/2309.09032/) (2023)
- [Near Optimal Heteroscedastic Regression with Symbiotic Learning](https://mlsanity.com/paper/2306.14288/) (2023)
- [Optimal sparse phase retrieval via a quasi-Bayesian approach](https://mlsanity.com/paper/2504.09509/) (2025)
- [Total Least Squares Phase Retrieval](https://mlsanity.com/paper/2102.00927/) (2021)
- [Robust High-Dimensional Regression with Coefficient Thresholding and its Application to Imaging Data Analysis](https://mlsanity.com/paper/2109.14856/) (2021)
- [Poisson-Gaussian Holographic Phase Retrieval with Score-based Image Prior](https://mlsanity.com/paper/2305.07712/) (2023)
- [Recovering Simultaneously Structured Data via Non-Convex Iteratively Reweighted Least Squares](https://mlsanity.com/paper/2306.04961/) (2023)
- [Mode-wise Principal Subspace Pursuit and Matrix Spiked Covariance Model](https://mlsanity.com/paper/2307.00575/) (2023)
- [Computationally efficient reductions between some statistical models](https://mlsanity.com/paper/2402.07717/) (2024)
- [Low Complexity Regularized Phase Retrieval](https://mlsanity.com/paper/2407.16413/) (2024)
- [Smoothed Robust Phase Retrieval](https://mlsanity.com/paper/2409.01570/) (2024)
- [AltGDmin: Alternating GD and Minimization for Partly-Decoupled (Federated) Optimization](https://mlsanity.com/paper/2504.14741/) (2025)
- [Phase transition of descending phase retrieval algorithms](https://mlsanity.com/paper/2506.18275/) (2025)
- [Optimal spectral initializers impact on phase retrieval phase transitions -- an RDT view](https://mlsanity.com/paper/2506.18279/) (2025)
- [Phase retrieval with rank d measurements -- descending algorithms phase transitions](https://mlsanity.com/paper/2506.18282/) (2025)

## 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.