# Sparse Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms

arXiv: https://arxiv.org/abs/1311.2745 · HTML: https://mlsanity.com/paper/1311.2745/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Nov 2013.

## Built on by (most cited first, 8 of 8)

- [Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priors](https://mlsanity.com/paper/2309.09032/) (2023)
- [Sparse Multi-Reference Alignment : Phase Retrieval, Uniform Uncertainty Principles and the Beltway Problem](https://mlsanity.com/paper/2106.12996/) (2021)
- [Recovering Simultaneously Structured Data via Non-Convex Iteratively Reweighted Least Squares](https://mlsanity.com/paper/2306.04961/) (2023)
- [Low Complexity Regularized Phase Retrieval](https://mlsanity.com/paper/2407.16413/) (2024)
- [PtyGenography: using generative models for regularization of the phase retrieval problem](https://mlsanity.com/paper/2502.01338/) (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)

The problem of signal recovery from its Fourier transform magnitude is of paramount importance in various fields of engineering and has been around for over 100 years. Due to the absence of phase information, some form of additional information is required in order to be able to uniquely identify the signal of interest. In this work, we focus our attention on discrete-time sparse signals (of length n). We first show that, if the DFT dimension is greater than or equal to 2n, almost all signals with em aperiodic support can be uniquely identified by their Fourier transform magnitude (up to time-shift, conjugate-flip and global phase). Then, we develop an efficient Two-stage Sparse Phase Retrieval algorithm (TSPR), which involves: (i) identifying the support, i.e., the locations of the non-zero components, of the signal using a combinatorial algorithm (ii) identifying the signal values in the support using a convex algorithm. We show that TSPR can em provably recover most O(n1/2-eps)-sparse signals (up to a time-shift, conjugate-flip and global phase). We also show that, for most O(n1/4-eps)-sparse signals, the recovery is em robust in the presence of measurement noise. Numerical experiments complement our theoretical analysis and verify the effectiveness of TSPR.