# Solving Complex Quadratic Systems with Full-Rank Random Matrices

## TL;DR

This paper demonstrates that complex quadratic systems with full-rank random matrices can be efficiently solved using spectral initialization and Wirtinger flow, extending phase retrieval techniques to more general and practical measurement models.

## Contribution

It introduces a novel approach to solve full-rank complex quadratic systems with rotation-invariant sub-Gaussian matrices, broadening applicability beyond rank-1 cases.

## Key findings

- High-probability success of spectral initialization and Wirtinger flow
- Effective solution for complex full-rank measurement models
- Numerical experiments confirm theoretical results

## Abstract

We tackle the problem of recovering a complex signal $\boldsymbol x\in\mathbb{C}^n$ from quadratic measurements of the form $y_i=\boldsymbol x^*\boldsymbol A_i\boldsymbol x$, where $\boldsymbol A_i$ 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.