Characterizations of matrix and operator-valued Φ-entropies and operator Efron–Stein inequalities

摘要

We derive new characterizations of the matrix Φ-entropy functionals introduced in Chen & Tropp (Chen, Tropp 2014 Electron. J. Prob. >19, 1–30. ()). 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.