4 research outputs found
Solving Complex Quadratic Systems with Full-Rank Random Matrices
We tackle the problem of recovering a complex signal from quadratic measurements of the form , where is a full-rank,
complex random measurement matrix whose entries are generated from a
rotation-invariant sub-Gaussian distribution. We formulate it as the
minimization of a nonconvex loss. This problem is related to the well
understood phase retrieval problem where the measurement matrix is a rank-1
positive semidefinite matrix. Here we study the general full-rank case which
models a number of key applications such as molecular geometry recovery from
distance distributions and compound measurements in phaseless diffractive
imaging. Most prior works either address the rank-1 case or focus on real
measurements. The several papers that address the full-rank complex case adopt
the computationally-demanding semidefinite relaxation approach. In this paper
we prove that the general class of problems with rotation-invariant
sub-Gaussian measurement models can be efficiently solved with high probability
via the standard framework comprising a spectral initialization followed by
iterative Wirtinger flow updates on a nonconvex loss. Numerical experiments on
simulated data corroborate our theoretical analysis.Comment: This updated version of the manuscript addresses several important
issues in the initial arXiv submissio
On the Sample Complexity and Optimization Landscape for Quadratic Feasibility Problems
We consider the problem of recovering a complex vector from quadratic measurements . This problem, known as quadratic feasibility,
encompasses the well known phase retrieval problem and has applications in a
wide range of important areas including power system state estimation and x-ray
crystallography. In general, not only is the the quadratic feasibility problem
NP-hard to solve, but it may in fact be unidentifiable. In this paper, we
establish conditions under which this problem becomes {identifiable}, and
further prove isometry properties in the case when the matrices
are Hermitian matrices sampled from a complex Gaussian
distribution. Moreover, we explore a nonconvex {optimization} formulation of
this problem, and establish salient features of the associated optimization
landscape that enables gradient algorithms with an arbitrary initialization to
converge to a \emph{globally optimal} point with a high probability. Our
results also reveal sample complexity requirements for successfully identifying
a feasible solution in these contexts.Comment: 21 page