Universality of Linearized Message Passing for Phase Retrieval with Structured Sensing Matrices

TL;DR

This paper demonstrates that the state evolution of linearized message passing algorithms for phase retrieval remains universal across different structured sensing matrices, including Hadamard-Walsh matrices, under certain conditions.

Contribution

It proves the universality of state evolution for a class of linearized message passing algorithms with structured sensing matrices in phase retrieval.

Findings

State evolution holds for Haar-distributed orthogonal matrices.

State evolution also applies to Hadamard-Walsh sub-sampled matrices.

Universality is valid when the signal has a Gaussian prior.

Abstract

In the phase retrieval problem one seeks to recover an unknown $n$ dimensional signal vector $\mathbf{x}$ from $m$ measurements of the form $y_i = |(\mathbf{A} \mathbf{x})_i|$, where $\mathbf{A}$ denotes the sensing matrix. Many algorithms for this problem are based on approximate message passing. For these algorithms, it is known that if the sensing matrix $\mathbf{A}$ is generated by sub-sampling $n$ columns of a uniformly random (i.e., Haar distributed) orthogonal matrix, in the high dimensional asymptotic regime ($m,n \rightarrow \infty, n/m \rightarrow \kappa$), the dynamics of the algorithm are given by a deterministic recursion known as the state evolution. For a special class of linearized message-passing algorithms, we show that the state evolution is universal: it continues to hold even when $\mathbf{A}$ is generated by randomly sub-sampling columns of the Hadamard-Walsh matrix, provided the signal is drawn from a Gaussian prior.