## 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→→ℝ^{m} are injective, with α(x)=(|<x,f_{k}>|)_{k}^{m}=1 and β(x)=(|<x,f_{k}>|^{2})_{k}^{m}=1, where {f_{1},...,_{fm}} is a frame for a Hilbert space H and H=H/T^{1}, then α is bi-Lipschitz with respect to the class of "natural metrics" D_{p}(x,y)=min_{φ}||x-e^{iφ}y||_{p}, whereas β is bi-Lipschitz with respect to the class of matrix-norm induced metrics d_{p}(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:ℝ^{m}→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