Ramon van Handel

Sharp matrix concentration inequalities

What does the spectrum of a random matrix look like when we make no
assumption whatsoever about the covariance pattern of its entries? It may
appear hopeless that anything useful can be said at this level of
generality. Nonetheless, a set of tools known as "matrix concentration
inequalities" makes it possible to estimate at least the spectral norm of
very general random matrices up to logarithmic factors in the dimension.
On the other hand, it is well known that these inequalities fail to yield
sharp results for even the simplest random matrix models.

In this talk I will describe a powerful new class of matrix concentration
inequalities that achieve optimal results in many situations that are
outside the reach of classical methods. Our results are easily applicable
in concrete examples, and yield detailed nonasymptotic information on the
full spectrum of essentially arbitrarily structured random matrices. These
new inequalities arise from an unexpected phenomenon: the spectrum of
random matrices is accurately captured by certain predictions of free
probability theory under surprisingly minimal assumptions. Our proofs
quantify the notion that it costs little to be free.

The talk is based on joint works with Afonso Bandeira and March
Boedihardjo, and with Tatiana Brailovskaya. No prior background will be
assumed.