Kaczmarz Method for Solving Quadratic Equations

Abstract

Estimating low-rank positive-semidefinite (PSD) matrices from symmetric rank-one measurements is of great importance in many applications, such as high-dimensional data processing, quantum state tomography, and phase retrieval. When the rank is known a priori , this problem can be regarded as solving a system of quadratic equations of a low-dimensional subspace. The authors develop a fast iterative algorithm based on an adaptation of the Kaczmarz method, which is traditionally used for solving overdetermined linear systems. In particular, the authors characterize the dynamics of the algorithm when the measurement vectors are composed of standard Gaussian entries in the online setting. Numerical simulations demonstrate the compelling performance of the proposed algorithm.