The Paulsen Problem Made Simple

Authors Linus Hamilton, Ankur Moitra

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Linus Hamilton
  • Massachusetts Institute of Technology, 77 Massachusetts Ave, USA
Ankur Moitra
  • Massachusetts Institute of Technology, 77 Massachusetts Ave, USA

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Linus Hamilton and Ankur Moitra. The Paulsen Problem Made Simple. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 41:1-41:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


The Paulsen problem is a basic problem in operator theory that was resolved in a recent tour-de-force work of Kwok, Lau, Lee and Ramachandran. In particular, they showed that every epsilon-nearly equal norm Parseval frame in d dimensions is within squared distance O(epsilon d^{13/2}) of an equal norm Parseval frame. We give a dramatically simpler proof based on the notion of radial isotropic position, and along the way show an improved bound of O(epsilon d^2).

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • radial isotropic position
  • operator scaling
  • Paulsen problem


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