On Lipschitz analysis and Lipschitz synthesis for the phase retrieval problem

We prove two results with regard to reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”). First we show that phase retrievable nonlinear maps are bi-Lipschitz with respect to appropriate metrics on the quotient space. Specifically, if nonlinear analysis maps α,β:Hˆ→Rm are injective, with α(x)=(|〈x,fk〉|)k=1m and β(x)=(|〈x,fk〉|2)k=1m, where {f1,…,fm}{f1,…,fm} is a frame for a Hilbert space H   and Hˆ=H/T1, then α   is bi-Lipschitz with respect to the class of “natural metrics” Dp(x,y)=minφ⁡‖x−eiφy‖pDp(x,y)=minφ⁡‖x−eiφy‖p, whereas β   is bi-Lipschitz with respect to the class of matrix-norm induced metrics dp(x,y)=‖xx⁎−yy⁎‖pdp(x,y)=‖xx⁎−yy⁎‖p. Second we prove that reconstruction can be performed using Lipschitz continuous maps. That is, there exist left inverse maps (synthesis maps) ω,ψ:Rm→Hˆ of α and β respectively, that are Lipschitz continuous with respect to appropriate metrics. Additionally, we obtain the Lipschitz constants of ω and ψ in terms of the lower Lipschitz constants of α and β, respectively. Surprisingly, the increase in both Lipschitz constants is a relatively small factor, independent of the space dimension or the frame redundancy.