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-eiφy||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 language | English (US) |
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Pages (from-to) | 152-181 |
Number of pages | 30 |
Journal | Linear Algebra and Its Applications |
Volume | 496 |
DOIs | |
State | Published - 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.
Publisher Copyright:
© 2016 Elsevier Inc. All rights reserved.
Keywords
- Frames
- Lipschitz maps
- Phase retrieval
- Stability