Phase retrieval using random cubatures and fusion frames of positive semidefinite matrices

TL;DR

This paper generalizes phase retrieval to reconstruct symmetric rank-1 matrices using random cubatures and fusion frames of positive semidefinite matrices, enabling recovery via semidefinite programming.

Contribution

It introduces random cubatures for polynomial spaces and demonstrates their effectiveness in phase retrieval of symmetric rank-1 matrices.

Findings

Reconstruction achieved with high probability using random cubatures.

Provides a new framework for phase retrieval involving fusion frames.

Solves the problem via semidefinite programming feasibility.

Abstract

As a generalization of the standard phase retrieval problem, we seek to reconstruct symmetric rank-1 matrices from inner products with subclasses of positive semidefinite matrices. For such subclasses, we introduce random cubatures for spaces of multivariate polynomials based on moment conditions. The inner products with samples from sufficiently strong random cubatures allow the reconstruction of symmetric rank-1 matrices with a decent probability by solving the feasibility problem of a semidefinite program.