# A Smoothing Newton Method for Rank-one Matrix Recovery

arXiv: https://arxiv.org/abs/2507.23017 · HTML: https://mlsanity.com/paper/2507.23017/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 33 references that are in the map.
- First posted on arXiv 30 Jul 2025.

## Builds on (most cited first, 17 of 17)

- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution](https://mlsanity.com/paper/1711.10467/) (2017)
- [Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems](https://mlsanity.com/paper/1505.05114/) (2015)
- [Scalable Semidefinite Programming](https://mlsanity.com/paper/1912.02949/) (2019)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming](https://mlsanity.com/paper/1310.0807/) (2013)
- [ROP: Matrix recovery via rank-one projections](https://mlsanity.com/paper/1310.5791/) (2013)
- [Globally-convergent Iteratively Reweighted Least Squares for Robust Regression Problems](https://mlsanity.com/paper/2006.14211/) (2020)
- [Global Linear and Local Superlinear Convergence of IRLS for Non-Smooth Robust Regression](https://mlsanity.com/paper/2208.11846/) (2022)
- [Critical Points and Convergence Analysis of Generative Deep Linear Networks Trained with Bures-Wasserstein Loss](https://mlsanity.com/paper/2303.03027/) (2023)
- [Bures-Wasserstein Barycenters and Low-Rank Matrix Recovery](https://mlsanity.com/paper/2210.14671/) (2022)
- [Global Convergence of Iteratively Reweighted Least Squares for Robust Subspace Recovery](https://mlsanity.com/paper/2506.20533/) (2025)
- [Acceleration and Implicit Regularization in Gaussian Phase Retrieval](https://mlsanity.com/paper/2311.12888/) (2023)

## Abstract (arXiv)

We consider the phase retrieval problem, which involves recovering a rank-one positive semidefinite matrix from rank-one measurements. A recently proposed algorithm based on Bures-Wasserstein gradient descent (BWGD) exhibits superlinear convergence, but it is unstable, and existing theory can only prove local linear convergence for higher rank matrix recovery. We resolve this gap by revealing that BWGD implements Newton's method with a nonsmooth and nonconvex objective. We develop a smoothing framework that regularizes the objective, enabling a stable method with rigorous superlinear convergence guarantees. Experiments on synthetic data demonstrate this superior stability while maintaining fast convergence.