Phase Retrieval using Lipschitz Continuous Maps

TL;DR

This paper proves that phase retrieval from frame coefficients can be achieved using Lipschitz continuous maps, providing bounds on the inverse map's Lipschitz constant independent of space dimension or frame redundancy.

Contribution

It establishes the existence of a Lipschitz continuous inverse map for phase retrieval using frames, with explicit Lipschitz bounds.

Findings

Existence of a Lipschitz continuous inverse map for phase retrieval.

Lipschitz constant of the inverse depends on the analysis map's lower Lipschitz constant.

Lipschitz constant increase is independent of space dimension or frame redundancy.

Abstract

In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map $\alpha:{\mathcal H}\rightarrow\mathbb{R}^m$ is injective, with $(\alpha(x))_k=|<x,f_k>|^2$, where $\{f_1,\ldots,f_m\}$ is a frame for the Hilbert space ${\mathcal H}$, then there exists a left inverse map $\omega:\mathbb{R}^m\rightarrow {\mathcal H}$ that is Lipschitz continuous. Additionally we obtain the Lipschitz constant of this inverse map in terms of the lower Lipschitz constant of $\alpha$. Surprisingly the increase in Lipschitz constant is independent of the space dimension or frame redundancy.