The Paulsen problem made simple

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 $O(\epsilon d^2)$.

[1]  Avi Wigderson,et al.  Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via Operator Scaling , 2016, Geometric and Functional Analysis.

[2]  Richard G. Lynch,et al.  A brief introduction to Hilbert space frame theory and its applications , 2015, 1509.07347.

[3]  F. Barthe On a reverse form of the Brascamp-Lieb inequality , 1997, math/9705210.

[4]  Avi Wigderson,et al.  A Deterministic Polynomial Time Algorithm for Non-commutative Rational Identity Testing , 2015, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[5]  K. Ball Volumes of sections of cubes and related problems , 1989 .

[6]  Avi Wigderson,et al.  Superquadratic Lower Bound for 3-Query Locally Correctable Codes over the Reals , 2017, Theory Comput..

[7]  Peter G. Casazza,et al.  The Paulsen Problem in Operator Theory , 2011, 1102.2344.

[8]  Yin Tat Lee,et al.  The Paulsen problem, continuous operator scaling, and smoothed analysis , 2017, STOC.

[9]  Bernhard G. Bodmann,et al.  The road to equal-norm Parseval frames , 2010 .

[10]  Avi Wigderson,et al.  Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via Operator Scaling , 2018 .

[11]  Peter G. Casazza,et al.  Auto-tuning unit norm frames , 2010, 1009.5562.

[12]  T. Tao,et al.  The Brascamp–Lieb Inequalities: Finiteness, Structure and Extremals , 2005, math/0505065.

[13]  Jürgen Forster,et al.  A linear lower bound on the unbounded error probabilistic communication complexity , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[14]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[15]  Peter G. Casazza,et al.  The Kadison–Singer and Paulsen Problems in Finite Frame Theory , 2013 .

[16]  Leonid Gurvits,et al.  Classical complexity and quantum entanglement , 2004, J. Comput. Syst. Sci..

[17]  Ankur Moitra,et al.  Algorithms and Hardness for Robust Subspace Recovery , 2012, COLT.