User-Friendly Tail Bounds for Sums of Random Matrices

Joel Tropp - Caltech - Dept. of Computing & Mathematical Sciences
Sep 24 2013 - 11:00am
Event type: 
110 Fine Hall

Random matrices have come to play a significant role in computational mathematics and statistics.
Established methods from random matrix theory have led to striking advances in these areas, but
ongoing research has generated difficult questions that cannot be addressed without new tools.
The purpose of this talk is to introduce some recent techniques, collectively called matrix
concentration inequalities, that can simplify the study of many types of random matrices. These
results parallel classical tail bounds for scalar random variables, such as the Bernstein
inequality, but they apply directly to matrices. In particular, matrix concentration inequalities
can be used to control the spectral norm of a sum of independent random matrices by harnessing
basic properties of the summands. Many variants and extensions are now available, and the outlines
of a larger theory are starting to emerge.
These new techniques have already led to advances in many areas, including partial covariance
estimation, randomized schemes for low-rank matrix decomposition, relaxation and rounding
methods for combinatorial optimization, construction of maps for dimensionality reduction,
techniques for subsampling large matrices, analysis of sparse approximation algorithms,
and many others.