# Matrix measure concentration inequalities and bounds

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, $\mathbb{E}\|\mathbf{X}\| \leq \inf _{p \geq 1}\left(\mathbb{E}\|\mathbf{X}\|_{S_{p}}^{p}\right)^{1 / p}.$

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

Ahlswede, R., and A. Winter. 2002. IEEE Transactions on Information Theory 48 (3): 569–79.
Bach, Francis R. 2013. In COLT, 30:185–209.
Bishop, Adrian N., Pierre Del Moral, and Angèle Niclas. 2018. Foundations and Trends® in Machine Learning 11 (2): 97–218.
Boucheron, Stéphane, Gábor Lugosi, and Pascal Massart. 2013. Concentration Inequalities: A Nonasymptotic Theory of Independence. 1st ed. Oxford: Oxford University Press.
———. n.d. Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press.
Bousquet, Olivier, Ulrike von Luxburg, and Gunnar Rtsch. 2004. 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.
Brault, Romain, Florence d’Alché-Buc, and Markus Heinonen. 2016. In Proceedings of The 8th Asian Conference on Machine Learning, 110–25.
Candès, Emmanuel J., and Benjamin Recht. 2009. Foundations of Computational Mathematics 9 (6): 717–72.
Geer, Sara van de. 2014. arXiv:1409.8557 [Math, Stat], September.
Gross, D. 2011. IEEE Transactions on Information Theory 57 (3): 1548–66.
Gross, David, Yi-Kai Liu, Steven T. Flammia, Stephen Becker, and Jens Eisert. 2010. Physical Review Letters 105 (15).
Handel, Ramon van. 2017. In 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.
Mahoney, Michael W. 2016. arXiv Preprint arXiv:1608.04845.
Marton, Katalin. 2015. arXiv:1507.02803 [Math], July.
Massart, Pascal. 2007. 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.
Mohri, Mehryar, Afshin Rostamizadeh, and Ameet Talwalkar. 2018. Foundations of Machine Learning. Second edition. Adaptive Computation and Machine Learning. Cambridge, Massachusetts: The MIT Press.
Oliveira, Roberto Imbuzeiro. 2010. Journal of Combinatorics 1 (3): 285–306.
Paulin, Daniel, Lester Mackey, and Joel A. Tropp. 2016. The Annals of Probability 44 (5): 3431–73.
Stam, A. J. 1982. Journal of Applied Probability 19 (1): 221–28.
Tropp, Joel. 2019. Matrix Concentration & Computational Linear Algebra / ENS Short Course.
Tropp, Joel A. 2015. An Introduction to Matrix Concentration Inequalities.
Vershynin, Roman. 2018. High-Dimensional Probability: An Introduction with Applications in Data Science. 1st ed. Cambridge University Press.
Woodruff, David P. 2014. Sketching as a Tool for Numerical Linear Algebra. Foundations and Trends in Theoretical Computer Science 1.0. Now Publishers.

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