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Dvir, Zeev; Saraf, Shubhangi; Wigderson, Avi

doi:10.4086/toc.2017.v013a011

Cited by 7 publications

(1 citation statement)

References 13 publications

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“…The use of such vectors can be traced back to Fritz John's seminal work [Joh48] which led to the notion of the John ellipsoid of a convex body. Since then radial isotropy for vectors has found applications to numerous other areas such as the complexity of unbounded error probabilistic communication protocols [For02]; algorithmic and optimization aspects of Brascamp-Lieb inequalities [BCCT08]; superquadratic lower bounds for 3-query correctable codes [DSW17]; and algorithmic aspects of point location in high-dimensional arrangements of hyperplanes [KLM18].…”

Section: Background and Motivationmentioning

confidence: 99%

A Quiver Invariant Theoretic Approach to Radial Isotropy and the Paulsen Problem for Matrix Frames

Chindris

Ismaeel

2022

SIAM J. Appl. Algebra Geometry

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In this dissertation, we view matrix frames as representations of quivers and study them within the general framework of Quiver Invariant Theory. We are particularly interested in radial isotropic and Parseval matrix frames. Using methods from Quiver Invariant Theory [CD21], we first prove a far-reaching generalization of Barthe's Theorem [Bar98] on vectors in radial isotropic position to the case of matrix frames (see Theorems 5.13(3) and 4.12). With this tool at our disposal, we generalize the Paulsen problem from frames (of vectors) to frames of matrices of arbitrary rank and size extending Hamilton-Moitra's upper bound [HM18]. Specifically, we show in Theorem 5.20 that for any given ε-nearly equal-norm Parseval frame F of n matrices with d rows there exists an equal-norm Parseval frame W of n matrices with d rows such that dist 2 pF, Wq ď 46εd 2 . Finally, in Theorem 5.28 we address the constructive aspects of transforming a matrix frame into radial isotropic position which extend those in [Bar98, AKS20]. v

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Section: Background and Motivationmentioning

confidence: 99%

A Quiver Invariant Theoretic Approach to Radial Isotropy and the Paulsen Problem for Matrix Frames

Chindris

Ismaeel

2022

SIAM J. Appl. Algebra Geometry

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The Paulsen problem made simple

Hamilton

Moitra

2021

Isr. J. Math.

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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

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