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A Smoothing Newton Method for Rank-one Matrix Recovery

arXiv 2507.23017 ↗First posted 30 Jul 2025stat.ML cs.LG math.OCcited by 0

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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.

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.

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References · 17

  1. 2014Convex Optimization: Algorithms and Complexitycited by 575
  2. 2019Implicit Regularization in Deep Matrix Factorizationcited by 405
  3. 2018Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overviewcited by 160
  4. 2014Phase Retrieval via Wirtinger Flow: Theory and Algorithmscited by 125
  5. 2022Variational inference via Wasserstein gradient flowscited by 85
  6. 2017Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolutioncited by 63
  7. 2015Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systemscited by 50
  8. 2019Scalable Semidefinite Programmingcited by 36
  9. 2023Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Spacecited by 36
  10. 2013Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programmingcited by 20
  11. 2013ROP: Matrix recovery via rank-one projectionscited by 18
  12. 2020Globally-convergent Iteratively Reweighted Least Squares for Robust Regression Problemscited by 15
5 more
  1. 2022Global Linear and Local Superlinear Convergence of IRLS for Non-Smooth Robust Regressioncited by 10
  2. 2023Critical Points and Convergence Analysis of Generative Deep Linear Networks Trained with Bures-Wasserstein Losscited by 7
  3. 2022Bures-Wasserstein Barycenters and Low-Rank Matrix Recoverycited by 5
  4. 2025Global Convergence of Iteratively Reweighted Least Squares for Robust Subspace Recoverycited by 3
  5. 2023Acceleration and Implicit Regularization in Gaussian Phase Retrievalcited by 1

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