Recovering rank-one matrices via rank-r matrices relaxation

TL;DR

This paper introduces a relaxation method combined with nonconvex alternating minimization to recover rank-one matrices from measurements, validated through empirical studies with Gaussian matrices.

Contribution

It proposes a novel relaxation approach integrated with nonconvex methods for rank-one matrix recovery, enhancing phase retrieval techniques.

Findings

Effective recovery with Gaussian random matrices

Singular vectors approximate the unknown signal

Relaxation improves initialization for nonconvex algorithms

Abstract

PhaseLift, proposed by E.J. Cand\`{e}s et al., is one convex relaxation approach for phase retrieval. The relaxation enlarges the solution set from rank one matrices to positive semidefinite matrices. In this paper, a relaxation is employed to nonconvex alternating minimization methods to recover the rank-one matrices. A generic measurement matrix can be standardized to a matrix consisting of orthonormal columns. To recover the rank-one matrix, the standardized frames are used to select the matrix with the maximal leading eigenvalue among the rank-$r$ matrices. Empirical studies are conducted to validate the effectiveness of this relaxation approach. In the case of Gaussian random matrices with a sufficient number of nearly orthogonal sensing vectors, we show that the singular vector corresponding to the least singular value is close to the unknown signal, and thus it can be a good initialization for the nonconvex minimization algorithm.