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).
@InProceedings{hamilton_et_al:LIPIcs.ITCS.2019.41, author = {Hamilton, Linus and Moitra, Ankur}, title = {{The Paulsen Problem Made Simple}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {41:1--41:6}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.41}, URN = {urn:nbn:de:0030-drops-101347}, doi = {10.4230/LIPIcs.ITCS.2019.41}, annote = {Keywords: radial isotropic position, operator scaling, Paulsen problem} }
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