Phase retrieval for affine groups over prime fields

arXiv (Cornell University)

Abstract

We study phase retrieval for group frames arising from permutation representations, focusing on the action of the affine group of a finite field. We investigate various versions of the phase retrieval problem, including conjugate phase retrieval, sign retrieval, and matrix recovery. Our main result establishes that the canonical irreducible representation of the affine group $\mathbb{Z}_p \rtimes \mathbb{Z}_p^\ast$ (with $p$ prime), acting on the vectors in $\mathbb{C}^{p}$ with zero-sum, has the strongest retrieval property, allowing to reconstruct matrices from scalar products with a group orbit consisting of rank-one projections. We explicitly characterize the generating vectors that ensure this property, provide a linear matrix recovery algorithm and explicit examples of vectors that allow matrix recovery. We also comment on more general permutation representations.

Authors 3

1. David Bartusel Aachen Lehrstuhl für Geometrie und Analysis

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Geometrie und Analysis RWTH Aachen University D-52056 Aachen Germany

2. Hartmut Führ
3. Vignon Oussa Aachen Lehrstuhl für Geometrie und Analysis

   RWTH Aachen University · Bridgewater State University

   Affiliation as printed

   Department of Mathematics Bridgewater State University MA 02234 USA

   Lehrstuhl für Geometrie und Analysis RWTH Aachen University D-52056 Aachen Germany