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 - <p style="text-align: justify;">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.</p>