Abstract:
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]
