On Lipschitz Analysis and Lipschitz Synthesis for the Phase Retrieval Problem

TL;DR

This paper establishes that phase retrieval can be characterized and performed using Lipschitz continuous maps, providing bounds on reconstruction stability and inverse map Lipschitz constants.

Contribution

It proves that phase retrievability implies bi-Lipschitz properties of analysis maps and constructs Lipschitz continuous inverse maps for phase retrieval.

Findings

Phase retrievability implies bi-Lipschitz maps.

Existence of Lipschitz continuous inverse maps.

Lipschitz constants are bounded independently of space dimension.

Abstract

In this paper we prove two results regarding reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem"). First we show that phase retrievability as an algebraic property implies that nonlinear maps are bi-Lipschitz with respect to appropriate metrics on the quotient space. Second we prove that reconstruction can be performed using Lipschitz continuous maps. Specifically we show that when nonlinear analysis maps $\alpha,\beta:\hat{H}\rightarrow R^m$ are injective, with $\alpha(x)=(|\langle x,f_k\rangle |)_{k=1}^m$ and $\beta(x)=(|\langle x,f_k \rangle|^2)_{k=1}^m$, where $\{f_1,\ldots,f_m\}$ is a frame for a Hilbert space $H$ and $\hat{H}=H/T^1$, then $\alpha$ is bi-Lipschitz with respect to the class of "natural metrics" $D_p(x,y)= min_{\varphi} || x-e^{i\varphi}y {||}_p$, whereas $\beta$ is bi-Lipschitz with respect to the class of matrix-norm induced metrics $d_p(x,y)=|| xx^*-yy^*{||}_p$. Furthermore, there exist left inverse maps $\omega,\psi:R^m\rightarrow \hat{H}$ of $\alpha$ and $\beta$ respectively, that are Lipschitz continuous with respect to the appropriate metric. Additionally we obtain the Lipschitz constants of these inverse maps in terms of the lower Lipschitz constants of $\alpha$ and $\beta$. Surprisingly the increase in Lipschitz constant is a relatively small factor, independent of the space dimension or the frame redundancy.