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Volume 42, Issue 3
Solving Systems of Phaseless Equations via Riemannian Optimization with Optimal Sampling Complexity

Jianfeng Cai & Ke Wei

DOI: 10.4208/jcm.2207-m2021-0247

J. Comp. Math., 42 (2024), pp. 755-783.

Published online: 2024-04

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  • Abstract

A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations $|Ax|^2=y.$ The algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-1 positive semidefinite matrices. Theoretical recovery guarantee has been established for the truncated variant, showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of unknowns. Two key ingredients in the analysis are the restricted well conditioned property and the restricted weak correlation property of the associated truncated linear operator. Empirical evaluations show that our algorithms are competitive with other state-of-the-art first order nonconvex approaches with provable guarantees.

  • Keywords

Phaseless equations, Riemannian gradient descent, Manifold of rank-1 and positive semidefinite matrices, Optimal sampling complexity.

  • AMS Subject Headings

65F55, 65K05, 90C26, 90C30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-755, author = {Cai , Jianfeng and Wei , Ke}, title = {Solving Systems of Phaseless Equations via Riemannian Optimization with Optimal Sampling Complexity}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {3}, pages = {755--783}, abstract = {

A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations $|Ax|^2=y.$ The algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-1 positive semidefinite matrices. Theoretical recovery guarantee has been established for the truncated variant, showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of unknowns. Two key ingredients in the analysis are the restricted well conditioned property and the restricted weak correlation property of the associated truncated linear operator. Empirical evaluations show that our algorithms are competitive with other state-of-the-art first order nonconvex approaches with provable guarantees.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2207-m2021-0247}, url = {http://global-sci.org/intro/article_detail/jcm/23035.html} }
TY - JOUR T1 - Solving Systems of Phaseless Equations via Riemannian Optimization with Optimal Sampling Complexity AU - Cai , Jianfeng AU - Wei , Ke JO - Journal of Computational Mathematics VL - 3 SP - 755 EP - 783 PY - 2024 DA - 2024/04 SN - 42 DO - http://doi.org/10.4208/jcm.2207-m2021-0247 UR - https://global-sci.org/intro/article_detail/jcm/23035.html KW - Phaseless equations, Riemannian gradient descent, Manifold of rank-1 and positive semidefinite matrices, Optimal sampling complexity. AB -

A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations $|Ax|^2=y.$ The algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-1 positive semidefinite matrices. Theoretical recovery guarantee has been established for the truncated variant, showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of unknowns. Two key ingredients in the analysis are the restricted well conditioned property and the restricted weak correlation property of the associated truncated linear operator. Empirical evaluations show that our algorithms are competitive with other state-of-the-art first order nonconvex approaches with provable guarantees.

Jianfeng Cai & Ke Wei. (2024). Solving Systems of Phaseless Equations via Riemannian Optimization with Optimal Sampling Complexity. Journal of Computational Mathematics. 42 (3). 755-783. doi:10.4208/jcm.2207-m2021-0247
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