Characterisations of Matrix and Operator-Valued $Φ$-Entropies, and Operator Efron-Stein Inequalities
نویسندگان
چکیده
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.
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ورودعنوان ژورنال:
- Proceedings. Mathematical, physical, and engineering sciences
دوره 472 2187 شماره
صفحات -
تاریخ انتشار 2016