Concentration inequalities for matrix-valued random variables.

Recommended overviews are J. A. Tropp (2015); van Handel (2017); Vershynin (2018).

## Matrix Chernoff

J. A. Tropp (2015) summarises:

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.

Are these related?

Nikhil Srivastava’s Discrepancy, Graphs, and the Kadison-Singer Problem has an interesting example of bounds via discrepancy theory (and only indirectly probability). D. Gross (2011) is also readable, and gives results for matrices over the complex field.

## Matrix Chebychev

As discussed in, e.g. Paulin, Mackey, and Tropp (2016).

Let \(\mathbf{X} \in \mathbb{H}^{d}\) be a random matrix. For all \(t>0\) \[ \mathbb{P}\{\|\mathbf{X}\| \geq t\} \leq \inf _{p \geq 1} t^{-p} \cdot \mathbb{E}\|\mathbf{X}\|_{S_{p}}^{p} \] Furthermore, \[ \mathbb{E}\|\mathbf{X}\| \leq \inf _{p \geq 1}\left(\mathbb{E}\|\mathbf{X}\|_{S_{p}}^{p}\right)^{1 / p}. \]

## Matrix Bernstein

TBC.

## Matrix Efron-Stein

The “classical” Efron-Stein inequalities are simple. The Matrix ones, not so much

e.g. Paulin, Mackey, and Tropp (2016).

## Gaussian

Handy results from Vershynin (2018):

Takes \(X \sim N\left(0, I_{n}\right).\)

Show that, for any fixed vectors \(u, v \in \mathbb{R}^{n},\) we have \[ \mathbb{E}\langle X, u\rangle\langle X, v\rangle=\langle u, v\rangle \]

Given a vector \(u \in \mathbb{R}^{n}\), consider the random variable \(X_{u}:=\langle X, u\rangle .\)

Further, we know that \(X_{u} \sim N\left(0,\|u\|_{2}^{2}\right) .\) It follows that \[ \mathbb{E}\left[(X_{u}-X_{v})^2\right]^{1/2}=\|u-v\|_{2} \] for any fixed vectors \(u, v \in \mathbb{R}^{n} .\)

Grothendieck’s identity: For any fixed vectors \(u, v \in S^{n-1},\) we have \[ \mathbb{E} \operatorname{sign}X_{u} \operatorname{sign}X_{v}=\frac{2}{\pi} \arcsin \langle u, v\rangle. \]

## References

*IEEE Transactions on Information Theory*48 (3): 569–79.

*COLT*, 30:185–209.

*Foundations and Trends® in Machine Learning*11 (2): 97–218.

*Concentration Inequalities: A Nonasymptotic Theory of Independence*. 1st ed. Oxford: Oxford University Press.

*Concentration Inequalities: A Nonasymptotic Theory of Independence*. Oxford University Press.

*Advanced Lectures on Machine Learning: ML Summer Schools 2003, Canberra, Australia, February 2-14, 2003, T Bingen, Germany, August 4-16, 2003, Revised Lectures*. Springer.

*Proceedings of The 8th Asian Conference on Machine Learning*, 110–25.

*Foundations of Computational Mathematics*9 (6): 717–72.

*arXiv:1409.8557 [Math, Stat]*, September.

*IEEE Transactions on Information Theory*57 (3): 1548–66.

*Physical Review Letters*105 (15).

*Convexity and Concentration*, edited by Eric Carlen, Mokshay Madiman, and Elisabeth M. Werner, 107–56. The IMA Volumes in Mathematics and Its Applications. New York, NY: Springer.

*arXiv Preprint arXiv:1608.04845*.

*arXiv:1507.02803 [Math]*, July.

*Concentration Inequalities and Model Selection: Ecole d’Eté de Probabilités de Saint-Flour XXXIII - 2003*. Lecture Notes in Mathematics 1896. Berlin ; New York: Springer-Verlag.

*Foundations of Machine Learning*. Second edition. Adaptive Computation and Machine Learning. Cambridge, Massachusetts: The MIT Press.

*Journal of Combinatorics*1 (3): 285–306.

*The Annals of Probability*44 (5): 3431–73.

*Journal of Applied Probability*19 (1): 221–28.

*Matrix Concentration & Computational Linear Algebra / ENS Short Course*.

*An Introduction to Matrix Concentration Inequalities*.

*High-Dimensional Probability: An Introduction with Applications in Data Science*. 1st ed. Cambridge University Press.

*Sketching as a Tool for Numerical Linear Algebra*. Foundations and Trends in Theoretical Computer Science 1.0. Now Publishers.

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