Characterizations of matrix and operator-valued Φ-entropies, and operator Efron–Stein inequalities
Journal
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
Journal Volume
472
Journal Issue
2187
Date Issued
2016
Author(s)
Min-Hsiu Hsieh
Abstract
We derive new characterizations of the matrix Φ-entropy functionals introduced in Chen & Tropp (Chen, Tropp 2014 Electron. J. Prob.19, 1-30. (doi:10.1214/ejp.v19-2964)). These characterizations help us to better understand the properties of matrix Φ-entropies, and are a powerful tool for establishing matrix concentration inequalities for random matrices. Then, we propose an operator-valued generalization of matrix Φ-entropy functionals, and prove the subadditivity under Löwner partial ordering. Our results demonstrate that the subadditivity of operator-valued Φ-entropies is equivalent to the convexity. As an application, we derive the operator Efron-Stein inequality.
Subjects
Efron–Stein inequality; matrix concentration inequalities; Φ-entropy; Mathematical Physics; Mathematical Physics; Computer Science - Information Theory; Mathematics - Information Theory; Mathematics - Mathematical Physics; Mathematics - Probability; Quantum Physics
Publisher
ROYAL SOC
Description
21 pages. Text partially overlaps with arXiv:1506.06801. Accepted in
Proceedings of the Royal Society A: Mathematical, Physical & Engineering
Sciences
Proceedings of the Royal Society A: Mathematical, Physical & Engineering
Sciences
Type
journal article