# An Introduction to Matrix Concentration Inequalities

@article{Tropp2015AnIT, title={An Introduction to Matrix Concentration Inequalities}, author={Joel A. Tropp}, journal={ArXiv}, year={2015}, volume={abs/1501.01571} }

In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix concentration inequalities, research has advanced to the point where we can conquer many (formerly) challenging problems with a page or two of arithmetic. The aim of this monograph is to describe the most successful methods from this area along with some interesting…

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## References

SHOWING 1-10 OF 228 REFERENCES

Matrix concentration inequalities via the method of exchangeable pairs

- Mathematics
- 2014

This paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix. The analysis requires a matrix extension of the scalar…

The random paving property for uniformly bounded matrices

- Mathematics
- 2006

This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison-Singer problem. The result shows that every unit-norm matrix whose…

Concentration Inequalities

- MathematicsCOLT
- 2003

This text attempts to summarize some of the basic tools used in establishing concentration inequalities, which are at the heart of the mathematical analysis of various problems in machine learning and made it possible to derive new efficient algorithms.

Concentration Inequalities - A Nonasymptotic Theory of Independence

- MathematicsConcentration Inequalities
- 2013

Deep connections with isoperimetric problems are revealed whilst special attention is paid to applications to the supremum of empirical processes.

Sketching as a Tool for Numerical Linear Algebra

- Mathematics, Computer ScienceFound. Trends Theor. Comput. Sci.
- 2014

This survey highlights the recent advances in algorithms for numericallinear algebra that have come from the technique of linear sketching, and considers least squares as well as robust regression problems, low rank approximation, and graph sparsification.

A survey of certain trace inequalities

- Mathematics
- 1994

This paper concerns inequalities like TrA ≤ TrB, where A and B are certain Hermitian complex matrices and Tr stands for the trace. In most cases A and B will be exponential or logarithmic expressions…

Sums of random Hermitian matrices and an inequality by Rudelson

- Mathematics
- 2010

We give a new, elementary proof of a key inequality used by Rudelson in the derivation of his well-known bound for random sums of rank-one operators. Our approach is based on Ahlswede and Winter's…

User-Friendly Tail Bounds for Sums of Random Matrices

- MathematicsFound. Comput. Math.
- 2012

This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices and provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid.