Phase retrieval using alternating minimization in a batch setting

TL;DR

This paper introduces a modified alternating minimization algorithm for phase retrieval in a batch setting, proving it can recover signals with high probability when the number of measurements is proportional to n log^3 n.

Contribution

It proposes a new batch-based alternating minimization method and provides theoretical guarantees for signal recovery with fewer measurements than previous methods.

Findings

Recovery with high probability when m=O(n log^3 n)

Algorithm reduces the angle between estimate and true signal after each iteration

Improves understanding of phase retrieval with random initialization

Abstract

This paper considers the problem of phase retrieval, where the goal is to recover a signal $z\in C^n$ from the observations $y_i=|a_i^* z|$, $i=1,2,\cdots,m$. While many algorithms have been proposed, the alternating minimization algorithm has been one of the most commonly used methods, and it is very simple to implement. Current work has proved that when the observation vectors $\{a_i\}_{i=1}^m$ are sampled from a complex Gaussian distribution $N(0, I)$, it recovers the underlying signal with a good initialization when $m=O(n)$, or with random initialization when $m=O(n^2)$, and it conjectured that random initialization succeeds with $m=O(n)$. This work proposes a modified alternating minimization method in a batch setting, and proves that when $m=O(n\log^{3}n)$, the proposed algorithm with random initialization recovers the underlying signal with high probability. The proof is based on the observation that after each iteration of alternating minimization, with high probability, the angle between the estimated signal and the underlying signal is reduced.