Matrix concentration inequalities via the method of exchangeable pairs

We develop a theoretical framework for establishing concentration inequalities for non commuta- tive operators, focusing specically for the spectral norm of random matrices. Our work reposes on Stein's method of exchangeable pairs, as elaborated by Chatterjee, and it provides a very differ- ent approach to concentration than that provided by the classical large deviation argument, which relies strongly on commutativity. When applied to a sum of independent random matrices, our approach yields matrix general izations of the classical inequalities due to Hoeding, Bernstein, Khintchine, and Rosenthal. The same technique delivers bounds for sums of dependent random matrices and more general matrix valued functions of dependent random variables. [Joint work with Lester Mackey, Richard Chen, Brendan Farrell, and Joel Tropp]