Solving PhaseLift by low rank Riemannian optimization methods for complex semidefinite constraints

Authors

Wen Huang, K. A. Gallivan, Xiangxiong Zhang

Abstract

A framework, PhaseLift, was recently proposed to solve the phase retrieval problem. In this framework, the problem is solved by optimizing a cost function over the set of complex Hermitian positive semidefinite matrices. This approach to phase retrieval motivates a more general consideration of optimizing cost functions on semidefinite Hermitian matrices where the desired minimizers are known to have low rank. This paper considers an approach based on an alternative cost function defined on a union of appropriate manifolds. It is related to the original cost function in a manner that preserves the ability to find a global minimizer and is significantly more efficient computationally. A rank-based optimality condition for stationary points is given and optimization algorithms based on state-of-the-art Riemannian optimization and dynamically reducing rank are proposed. Empirical evaluations are performed using the PhaseLift problem. The new approach is shown to be an effective method of phase retrieval with computational efficiency increased substantially compared to the algorithm used in original PhaseLift paper. A preliminary version can be found in [HGZ16].

Key words

Riemannian Optimization; Low rank optimization; Complex optimization; Phase Retrieval; PhaseLift

Status

SIAM Journal on Scientific Computing, 39:5, pp. B840-B859, 2017.

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