Efron–Stein Inequalities for Random Matrices

Efron–stein Inequalities for Random Matrices By

Characterisations 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 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 gene...

Concentration inequalities using the entropy method

We investigate a new methodology, worked out by Ledoux and Mas-sart, to prove concentration-of-measure inequalities. The method is based on certain modified logarithmic Sobolev inequalities. We provide some very simple and general ready-to-use inequalities. One of these inequalities may be considered as an exponential version of the Efron-Stein inequality. The main purpose of this paper is to p...

New Characterizations of Matrix $Φ$-Entropies, Poincaré and Sobolev Inequalities and an Upper Bound to Holevo Quantity

We derive new characterizations of the matrix Φ-entropies introduced in [Electron. J. Probab., 19(20): 1–30, 2014 ]. These characterizations help to better understand the properties of matrix Φentropies, and are a powerful tool for establishing matrix concentration inequalities for matrix-valued functions of independent random variables. In particular, we use the subadditivity property to prove...

Singular value inequalities for positive semidefinite matrices

In this note‎, ‎we obtain some singular values inequalities for positive semidefinite matrices by using block matrix technique‎. ‎Our results are similar to some inequalities shown by Bhatia and Kittaneh in [Linear Algebra Appl‎. ‎308 (2000) 203-211] and [Linear Algebra Appl‎. ‎428 (2008) 2177-2191]‎.