Phase retrieval with random Gaussian sensing vectors by alternating projections

TL;DR

This paper proves that alternating projections can reliably recover a high-dimensional vector from phaseless measurements with random Gaussian sensing vectors, given a proper initialization, and supports the conjecture that no special initialization is needed.

Contribution

It establishes high-probability success of alternating projections for phase retrieval with Gaussian sensing vectors under suitable initialization, advancing theoretical understanding.

Findings

Success probability is high when m ≥ Cn with proper initialization

Numerical experiments support the conjecture without special initialization

Theoretical results apply to complex Gaussian sensing vectors

Abstract

We consider a phase retrieval problem, where we want to reconstruct a $n$-dimensional vector from its phaseless scalar products with $m$ sensing vectors, independently sampled from complex normal distributions. We show that, with a suitable initalization procedure, the classical algorithm of alternating projections succeeds with high probability when $m\geq Cn$, for some $C>0$. We conjecture that this result is still true when no special initialization procedure is used, and present numerical experiments that support this conjecture.