Phase retrieval in infinite-dimensional Hilbert spaces

TL;DR

This paper demonstrates that phase retrieval in infinite-dimensional Hilbert spaces lacks uniform stability, unlike the finite-dimensional case, and explores stability based on finite approximation quality.

Contribution

It establishes the fundamental instability of phase retrieval in infinite dimensions and provides stability results related to finite approximations.

Findings

Phase retrieval is never uniformly stable in infinite-dimensional Hilbert spaces.

Stability depends on how well signals are approximated by finite expansions.

Contrasts the stability properties with finite-dimensional phase retrieval.

Abstract

The main result of this paper states that phase retrieval in infinite-dimensional Hilbert spaces is never uniformly stable, in sharp contrast to the finite dimensional setting in which phase retrieval is always stable. This leads us to derive stability results for signals depending on how well they are approximated by finite expansions.