A Partial Derandomization of PhaseLift Using Spherical Designs

D. Gross, F. Krahmer, R. Kueng

Research output: Contribution to journalArticlepeer-review

61 Scopus citations

Abstract

The problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines over the last century. PhaseLift is a recently introduced algorithm for phase recovery that is computationally tractable, numerically stable, and comes with rigorous performance guarantees. PhaseLift is optimal in the sense that the number of amplitude measurements required for phase reconstruction scales linearly with the dimension of the signal. However, it specifically demands Gaussian random measurement vectors—a limitation that restricts practical utility and obscures the specific properties of measurement ensembles that enable phase retrieval. Here we present a partial derandomization of PhaseLift that only requires sampling from certain polynomial size vector configurations, called t-designs. Such configurations have been studied in algebraic combinatorics, coding theory, and quantum information. We prove reconstruction guarantees for a number of measurements that depends on the degree t of the design. If the degree is allowed to grow logarithmically with the dimension, the bounds become tight up to polylog-factors. Beyond the specific case of PhaseLift, this work highlights the utility of spherical designs for the derandomization of data recovery schemes.

Original languageEnglish
Pages (from-to)229-266
Number of pages38
JournalJournal of Fourier Analysis and Applications
Volume21
Issue number2
DOIs
StatePublished - Apr 2015
Externally publishedYes

Keywords

  • Non-commutative large deviation estimates
  • Phase retrieval
  • PhaseLift
  • Quantum information
  • Semidefinite relaxations of nonconvex quadratic programs
  • Spherical designs

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