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

Wen Huang (huwst08***at***gmail.com)
Kyle Gallivan (kgallivan***at***fsu.edu)
Xiangxiong Zhang (zhan1966***at***purdue.edu)

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.

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

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Citation: Universite catholique de Louvain, Tech. report UCL-INMA-2015.01, May 2016

Download: [PDF]

Entry Submitted: 05/18/2016
Entry Accepted: 05/18/2016
Entry Last Modified: 04/24/2017

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