# Hadamard Wirtinger Flow for Sparse Phase Retrieval

arXiv: https://arxiv.org/abs/2006.01065 · HTML: https://mlsanity.com/paper/2006.01065/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2020.

## Built on by (most cited first, 6 of 6)

- [More is Less: Inducing Sparsity via Overparameterization](https://mlsanity.com/paper/2112.11027/) (2021)
- [Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors](https://mlsanity.com/paper/2106.15358/) (2021)
- [Statistical-Computational Tradeoffs in Mixed Sparse Linear Regression](https://mlsanity.com/paper/2303.02118/) (2023)
- [Nearly Minimax-Optimal Rates for Noisy Sparse Phase Retrieval via Early-Stopped Mirror Descent](https://mlsanity.com/paper/2105.03678/) (2021)
- [Acceleration and Implicit Regularization in Gaussian Phase Retrieval](https://mlsanity.com/paper/2311.12888/) (2023)
- [Theoretical Analysis of Sparse Optimization with Reparameterization, Weight Decay, and Adaptive Learning Rate](https://mlsanity.com/paper/2605.25134/) (2026)

## Abstract (arXiv)

We consider the problem of reconstructing an n-dimensional k-sparse signal from a set of noiseless magnitude-only measurements. Formulating the problem as an unregularized empirical risk minimization task, we study the sample complexity performance of gradient descent with Hadamard parametrization, which we call Hadamard Wirtinger flow (HWF). Provided knowledge of the signal sparsity k, we prove that a single step of HWF is able to recover the support from k(x^*max)-2 (modulo logarithmic term) samples, where x^*max is the largest component of the signal in magnitude. This support recovery procedure can be used to initialize existing reconstruction methods and yields algorithms with total runtime proportional to the cost of reading the data and improved sample complexity, which is linear in k when the signal contains at least one large component. We numerically investigate the performance of HWF at convergence and show that, while not requiring any explicit form of regularization nor knowledge of k, HWF adapts to the signal sparsity and reconstructs sparse signals with fewer measurements than existing gradient based methods.