Information Theoretic Limits for Phase Retrieval with Subsampled Haar Sensing Matrices

TL;DR

This paper establishes the fundamental limits of signal recovery in phase retrieval using subsampled Haar sensing matrices, showing that below a certain measurement threshold, accurate recovery is impossible.

Contribution

It provides the first information-theoretic lower bound for phase retrieval with subsampled Haar matrices in high dimensions, matching known upper bounds.

Findings

If measurements are fewer than approximately twice the signal dimension, recovery is asymptotically impossible.

The lower bound on measurements is sharp, aligning with existing upper bounds for successful recovery.

Recovery becomes feasible when measurements exceed twice the signal dimension.

Abstract

We study information theoretic limits of recovering an unknown $n$ dimensional, complex signal vector $\mathbf{x}_\star$ with unit norm from $m$ magnitude-only measurements of the form $y_i = |(\mathbf{A} \mathbf{x}_\star)_i|^2, \; i = 1,2 \dots , m$, where $\mathbf{A}$ is the sensing matrix. This is known as the Phase Retrieval problem and models practical imaging systems where measuring the phase of the observations is difficult. Since in a number of applications, the sensing matrix has orthogonal columns, we model the sensing matrix as a subsampled Haar matrix formed by picking $n$ columns of a uniformly random $m \times m$ unitary matrix. We study this problem in the high dimensional asymptotic regime, where $m,n \rightarrow \infty$, while $m/n \rightarrow \delta$ with $\delta$ being a fixed number, and show that if $m < (2-o_n(1))\cdot n$, then any estimator is asymptotically orthogonal to the true signal vector $\mathbf{x}_\star$. This lower bound is sharp since when $m > (2+o_n(1)) \cdot n $, estimators that achieve a non trivial asymptotic correlation with the signal vector are known from previous works.