Phase Retrieval for Signals in Union of Subspaces

TL;DR

This paper introduces an algorithm for phase retrieval of signals in a union of subspaces, demonstrating that with proper initialization, a number of measurements proportional to the subspace dimension and the log of the number of subspaces suffices for accurate recovery.

Contribution

The paper proposes a novel joint estimation algorithm for phase and subspace support, with theoretical guarantees for signals in a union of low-dimensional subspaces.

Findings

Proper initialization enables accurate recovery with O(d + log R) measurements.

The algorithm effectively estimates phase and subspace support.

Simulation results confirm empirical performance.

Abstract

We consider the phase retrieval problem for signals that belong to a union of subspaces. We assume that amplitude measurements of the signal of length $n$ are observed after passing it through a random $m \times n$ measurement matrix. We also assume that the signal belongs to the span of a single $d$-dimensional subspace out of $R$ subspaces, where $d\ll n$. We assume the knowledge of all possible subspaces, but the true subspace of the signal is unknown. We present an algorithm that jointly estimates the phase of the measurements and the subspace support of the signal. We discuss theoretical guarantees on the recovery of signals and present simulation results to demonstrate the empirical performance of our proposed algorithm. Our main result suggests that if properly initialized, then $O(d+\log R)$ random measurements are sufficient for phase retrieval if the unknown signal belongs to the union of $R$ low-dimensional subspaces.