# Optimal sparse phase retrieval via a quasi-Bayesian approach

arXiv: https://arxiv.org/abs/2504.09509 · HTML: https://mlsanity.com/paper/2504.09509/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 53 references that are in the map.
- First posted on arXiv 13 Apr 2025.

## Builds on (most cited first, 21 of 21)

- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [A Primer on PAC-Bayesian Learning](https://mlsanity.com/paper/1901.05353/) (2019)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Phase Retrieval using Alternating Minimization](https://mlsanity.com/paper/1306.0160/) (2013)
- [Sparsity information and regularization in the horseshoe and other shrinkage priors](https://mlsanity.com/paper/1707.01694/) (2017)
- [Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow](https://mlsanity.com/paper/1506.03382/) (2015)
- [Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming](https://mlsanity.com/paper/1310.0807/) (2013)
- [Lasso Meets Horseshoe : A Survey](https://mlsanity.com/paper/1706.10179/) (2017)
- [A reduced-rank approach to predicting multiple binary responses through machine learning](https://mlsanity.com/paper/2306.05829/) (2023)
- [High-dimensional sparse classification using exponential weighting with empirical hinge loss](https://mlsanity.com/paper/2312.12952/) (2023)
- [Sparse Phase Retrieval via Truncated Amplitude Flow](https://mlsanity.com/paper/1611.07641/) (2016)
- [From bilinear regression to inductive matrix completion: a quasi-Bayesian analysis](https://mlsanity.com/paper/2210.15290/) (2022)
- [Misclassification bounds for PAC-Bayesian sparse deep learning](https://mlsanity.com/paper/2405.01304/) (2024)
- [A sparse PAC-Bayesian approach for high-dimensional quantile prediction](https://mlsanity.com/paper/2409.01687/) (2024)
- [Nearly Minimax-Optimal Rates for Noisy Sparse Phase Retrieval via Early-Stopped Mirror Descent](https://mlsanity.com/paper/2105.03678/) (2021)
- [Variable Selection Using Shrinkage Priors](https://mlsanity.com/paper/1503.04303/) (2015)
- [Compressive Phase Retrieval via Reweighted Amplitude Flow](https://mlsanity.com/paper/1712.02426/) (2017)

## Built on by (most cited first, 3 of 3)

- [Handling bounded response in high dimensions: a Horseshoe prior Bayesian Beta regression approach](https://mlsanity.com/paper/2505.22211/) (2025)
- [Kullback-Leibler excess risk bounds for exponential weighted aggregation in Generalized linear models](https://mlsanity.com/paper/2504.10171/) (2025)
- [Robust low-rank estimation with multiple binary responses using pairwise AUC loss](https://mlsanity.com/paper/2601.08618/) (2026)

## Abstract (arXiv)

This paper addresses the problem of sparse phase retrieval, a fundamental inverse problem in applied mathematics, physics, and engineering, where a signal need to be reconstructed using only the magnitude of its transformation while phase information remains inaccessible. Leveraging the inherent sparsity of many real-world signals, we introduce a novel sparse quasi-Bayesian approach and provide the first theoretical guarantees for such an approach. Specifically, we employ a scaled Student distribution as a continuous shrinkage prior to enforce sparsity and analyze the method using the PAC-Bayesian inequality framework. Our results establish that the proposed Bayesian estimator achieves minimax-optimal convergence rates under sub-exponential noise, matching those of state-of-the-art frequentist methods. To ensure computational feasibility, we develop an efficient Langevin Monte Carlo sampling algorithm. Through numerical experiments, we demonstrate that our method performs comparably to existing frequentist techniques, highlighting its potential as a principled alternative for sparse phase retrieval in noisy settings.