# Second-Order Matrix Concentration Inequalities

@article{Tropp2015SecondOrderMC,
title={Second-Order Matrix Concentration Inequalities},
author={Joel A. Tropp},
journal={arXiv: Probability},
year={2015}
}
• J. Tropp
• Published 22 April 2015
• Mathematics
• arXiv: Probability
Matrix concentration inequalities give bounds for the spectral-norm deviation of a random matrix from its expected value. These results have a weak dimensional dependence that is sometimes, but not always, necessary. This paper identifies one of the sources of the dimensional term and exploits this insight to develop sharper matrix concentration inequalities. In particular, this analysis delivers two refinements of the matrix Khintchine inequality that use information beyond the matrix variance… Expand
Moment inequalities for matrix-valued U-statistics of order 2
• Mathematics
• Electronic Journal of Probability
• 2019
We present Rosenthal-type moment inequalities for matrix-valued U-statistics of order 2. As a corollary, we obtain new matrix concentration inequalities for U-statistics. One of our main technicalExpand
Matrix Concentration for Products
• Mathematics
• Foundations of Computational Mathematics
• 2021
This paper develops nonasymptotic growth and concentration bounds for a product of independent random matrices. These results sharpen and generalize recent work of Henriksen-Ward, and they areExpand
An Introduction to Matrix Concentration Inequalities
• J. Tropp
• Computer Science, Mathematics
• Found. Trends Mach. Learn.
• 2015
The aim of this monograph is to describe the most successful methods from this area along with some interesting examples that these techniques can illuminate. Expand
Operator Norm Moment and Exponential Inequalities for Matrix U-statistics
• Mathematics
• 2018
We present a Rosenthal-type moment bound and a Bernstein-type exponential tail bound for order 1 degenerated matrix U-statistics in operator norm, with explicit factors on dimension dependencies. WeExpand
Matrix Concentration Inequalities and Free Probability
• Mathematics
• 2021
A central tool in the study of nonhomogeneous random matrices, the noncommutative Khintchine inequality of Lust-Piquard and Pisier, yields a nonasymptotic bound on the spectral norm of generalExpand
The Expected Norm of a Sum of Independent Random Matrices: An Elementary Approach
In contemporary applied and computational mathematics, a frequent challenge is to bound the expectation of the spectral norm of a sum of independent random matrices. This quantity is controlled byExpand
PR ] 1 3 A ug 2 02 1 MATRIX CONCENTRATION INEQUALITIES AND FREE PROBABILITY
A central tool in the study of nonhomogeneous random matrices, the noncommutative Khintchine inequality of Lust-Piquard and Pisier, yields a nonasymptotic bound on the spectral norm of generalExpand
Structured Random Matrices
Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area isExpand
Concentration Inequalities of Random Matrices and Solving Ptychography with a Convex Relaxation
Random matrix theory has seen rapid development in recent years. In particular, researchers have developed many non-asymptotic matrix concentration inequalities that parallel powerful scalarExpand
18.S096: Concentration Inequalities, Scalar and Matrix Versions
These are lecture notes not in final form and will be continuously edited and/or corrected (as I am sure it contains many typos). Please let me know if you find any typo/mistake. Also, I am postingExpand

#### References

SHOWING 1-10 OF 32 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 scalarExpand
An Introduction to Matrix Concentration Inequalities
• J. Tropp
• Computer Science, Mathematics
• Found. Trends Mach. Learn.
• 2015
The aim of this monograph is to describe the most successful methods from this area along with some interesting examples that these techniques can illuminate. Expand
User-Friendly Tail Bounds for Sums of Random Matrices
• J. Tropp
• Mathematics, Computer Science
• Found. 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. Expand
Subadditivity of Matrix phi-Entropy and Concentration of Random Matrices
• Computer Science, Mathematics
• ArXiv
• 2013
It is demonstrated that a class of entropy functionals defined for random matrices satisfy a subadditivity property, and several matrix concentration inequalities are derived as an application of this result. Expand
On the spectral norm of inhomogeneous random matrices
Let $X$ be a $d\times d$ symmetric random matrix with independent but non-identically distributed Gaussian entries. It has been conjectured by Latal{a} that the spectral norm of $X$ is always of theExpand
The Masked Sample Covariance Estimator: An Analysis via Matrix Concentration Inequalities
• Mathematics
• 2011
Covariance estimation becomes challenging in the regime where the number p of variables outstrips the number n of samples available to construct the estimate. One way to circumvent this problem is toExpand
Spectral Analysis of Large Dimensional Random Matrices
• Mathematics
• 2009
Wigner Matrices and Semicircular Law.- Sample Covariance Matrices and the Mar#x010D enko-Pastur Law.- Product of Two Random Matrices.- Limits of Extreme Eigenvalues.- Spectrum Separation.-Expand
Introduction to the non-asymptotic analysis of random matrices
• R. Vershynin
• Mathematics, Computer Science
• Compressed Sensing
• 2012
This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory, particularly for the problem of estimating covariance matrices in statistics and for validating probabilistic constructions of measurementMatrices in compressed sensing. Expand
Inequalities: A Journey into Linear Analysis
This book contains a wealth of inequalities used in linear analysis, and explains in detail how they are used. The book begins with Cauchy's inequality and ends with Grothendieck's inequality, inExpand
Optimal asymptotic bounds for spherical designs
• Mathematics
• 2010
In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$ there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constantExpand