### Sharp matrix concentration inequalities

Probability Seminar

8th October 2021, 3:30 pm – 4:30 pm

Fry Building, 2.04 (also on Zoom)

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.

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