Timezone: »
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 tutorial 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 lowrank 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.
Author Information
Joel A Tropp (Caltech)
Joel A. Tropp is Professor of Applied & Computational Mathematics at California Institute of Technology. He earned the Ph.D. degree in Computational Applied Mathematics from the University of Texas at Austin in 2004. Prof. Troppâ€™s work lies at the interface of applied mathematics, electrical engineering, computer science, and statistics. The bulk of this research concerns the theoretical and computational aspects of sparse approximation, compressive sampling, and randomized linear algebra. He has also worked extensively on the properties of structured random matrices. Prof. Tropp has received several major awards for young researchers, including the 2007 ONR Young Investigator Award and the 2008 Presidential Early Career Award for Scientists and Engineers. He is also winner of the 32nd annual award for Excellence in Teaching from the Associated Students of the California Institute of Technology.
More from the Same Authors

2017 Poster: FixedRank Approximation of a PositiveSemidefinite Matrix from Streaming Data »
Joel A Tropp · Alp Yurtsever · Madeleine Udell · Volkan Cevher 
2014 Poster: TimeData Tradeoffs by Aggressive Smoothing »
John J Bruer · Joel A Tropp · Volkan Cevher · Stephen Becker 
2012 Poster: Factoring nonnegative matrices with linear programs »
Benjamin Recht · Christopher Re · Joel A Tropp · Victor Bittorf 
2012 Spotlight: Factoring nonnegative matrices with linear programs »
Benjamin Recht · Christopher Re · Joel A Tropp · Victor Bittorf 
2010 Poster: Practical LargeScale Optimization for Maxnorm Regularization »
Jason D Lee · Benjamin Recht · Russ Salakhutdinov · Nati Srebro · Joel A Tropp