Solving complex quadratic systems with full-rank random matrices

Shuai Huang, Sidharth Gupta, Ivan Dokmanic

Research output: Contribution to journalArticlepeer-review


We tackle the problem of recovering a complex signal x ϵ Cn from quadratic measurements of the form yi= x∗Aix, where Ai 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.

Original languageEnglish (US)
Article number9146205
Pages (from-to)4782-4796
Number of pages15
JournalIEEE Transactions on Signal Processing
StatePublished - 2020


  • Complex quadratic equations
  • rotation invariance
  • spectral initialization
  • sub-Gaussian matrices

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering


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