The Paulsen problem made simple

Abstract

The Paulsen problem is a basic problem in operator theory that was resolved in a recent tourde-force work of Kwok, Lau, Lee and Ramachandran. In particular, they showed that every -nearly equal norm Parseval frame in d dimensions is within squared distance O( 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( d 2 ). ACM Subject Classification Theory of computation → Design and analysis of algorithmsKeywords and phrases radial isotropic position, operator scaling, Paulsen problem

On the Search for Tight Frames of Low Coherence

“…The set of finite unit norm frames is topologicaly connected, and an irreducible variety [18], but our experiment suggests that for a local minima of (1.10), there will be a tight frame close to it. This can perhaps be explained by the recently solved Paulsen Problem [46], which implies that for a nearly tight unit norm frame F , there exists a unit norm tight frame nearby since we have shown that a local minima of (1.10) is nearly tight. This suggests that step 3 will result in a nearby tight configuration if the configuration from step 2 is nearly tight as indicated in Theorem 6.4 for N large.…”

Paulsen and Projection Problems for Banach Spaces

“…In 2018, using radial isotropic position, Hamilton and Moitra [33,34] gave another proof of Paulsen problem which improved Theorem 1.23.…”

“…Theorem 1.24. [33,34] For any ε-nearly equal norm Parseval frame {τ j } n j=1 for R d , there is an equal norm Parseval frame {ω j } n j=1 for R d satisfying…”

Modular Paulsen Problem and Modular Projection Problem

“…In 2018, using radial isotropic position, Hamilton and Moitra [49,50] gave another proof of Paulsen problem which improved Theorem 1.24.…”