On Lipschitz analysis and Lipschitz synthesis for the phase retrieval problem

Radu Balan, Dongmian Zou

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

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:H→→ℝm are injective, with α(x)=(|<x,fk>|)km=1 and β(x)=(|<x,fk>|2)km=1, where {f1,...,fm} is a frame for a Hilbert space H and H=H/T1, then α is bi-Lipschitz with respect to the class of "natural metrics" Dp(x,y)=minφ||x-ey||p, whereas β is bi-Lipschitz with respect to the class of matrix-norm induced metrics dp(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→H 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.

Original languageEnglish (US)
Pages (from-to)152-181
Number of pages30
JournalLinear Algebra and Its Applications
Volume496
DOIs
StatePublished - May 1 2016

Bibliographical note

Funding Information:
The authors were supported in part by NSF grants DMS-1109498 and DMS-1413249 , and ARO grant W911NF-16-1-0008 . The first author acknowledges fruitful discussions with Krzysztof Nowak and Hugo Woerdeman (both from Drexel University) who pointed out several references, with Stanislav Minsker (Duke University) for pointing out [15] and [12] , and Vern Paulsen (University of Houston), Marcin Bownick (University of Oregon) and Friedrich Philipp (University of Berlin). We also thank the reviewers for their constructive comments and suggestions.

Keywords

  • Frames
  • Lipschitz maps
  • Phase retrieval
  • Stability

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