Characterizations of matrix and operator-valued Φ-entropies, and operator Efron-Stein inequalities.
We derive new characterizations of the matrix Φ-entropy functionals introduced in Chen & Tropp (Chen, Tropp 2014Electron. 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.
Graduate Institute of Communication Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, Republic of China; Faculty of Engineering and Information Technology, University of Technology Sydney, 15 Broadway, Ultimo, New South Wales 2007, Australia.
Hao-Chung
HC
Cheng
Faculty of Engineering and Information Technology , University of Technology Sydney , 15 Broadway, Ultimo, New South Wales 2007, Australia.
Min-Hsiu
MH
Hsieh
Efron–Stein inequality; matrix concentration inequalities; Φ-entropy