学术报告：Entropy method, concentration inequalities, and their matrix generalisation

中科院数学与系统科学研究院

数学研究所

学术报告会

报告人：Dr. Min-Hsiu Hsieh  （University of Technology, Sydney, Australia）

题  目： Entropy method, concentration inequalities, and their matrix generalisation

时  间：10.21 (星期三), 16:00--17:00 

地  点：数学院南楼N202室 

Abstract：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 Φ-entropies, 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 a Poincar\'e inequality for the matrix Φ-entropies. We also provide a new proof for the matrix Efron-Stein inequality. Furthermore, we derive logarithmic Sobolev inequalities for matrix-valued functions defined on Boolean hypercubes and with Gaussian distributions. Our proof relies on the powerful matrix Bonami-Beckner inequality. Finally, the Holevo quantity in quantum information theory is closely related to the matrix Φ-entropies. This allows us to upper bound the Holevo quantity of a classical-quantum ensemble that undergoes a special Markov evolution.