Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Phase retrieval in the general setting of continuous frames for Banach spaces

by R. Alaifari and P. Grohs

(Report number 2016-21)

Abstract
We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main result states that also in this framework, the problem of phase retrieval is never uniformly stable in infinite dimensions. On the other hand, we show weak stability of the problem. This complements recent work [9], where it has been shown that phase retrieval is always unstable for the setting of discrete frames in Hilbert spaces. In particular, our result implies that the stability properties cannot be improved by oversampling the underlying discrete frame. We generalize the notion of complement property (CP) to the setting of continuous frames for Banach spaces (over \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{K}=\mathbb{C}\)) and verify that it is a necessary condition for uniqueness of the phase retrieval problem; when \(\mathbb{K}=\mathbb{R}\) the CP is also sufficient for uniqueness. In our general setting, we also prove a conjecture posed by Bandeira et al. [5], which was originally formulated for finite-dimensional spaces: for the case \(\mathbb{K}=\mathbb{C}\) the strong complement property (SCP) is a necessary condition for stability. To prove our main result, we show that the SCP can never hold for frames of infinite-dimensional Banach spaces.

Keywords: Phase retrieval, continuous Banach frames, stability

@Techreport{AG16_658,
  author = {R. Alaifari and P. Grohs},
  title = {Phase retrieval in the general setting of continuous frames for Banach spaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-21},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-21.pdf },
  year = {2016}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).