research-article

A matrix expander Chernoff bound

Authors:

Nikhil SrivastavaAuthors Info & Claims

STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

Pages 1102 - 1114

https://doi.org/10.1145/3188745.3188890

Published: 20 June 2018 Publication History

Abstract

We prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a random walk on an expander, confirming a conjecture due to [Wigderson and Xiao 06]. Our proof is based on a new multi-matrix extension of the Golden-Thompson inequality which improves upon the inequality in [Sutter, Berta and Tomamichel 17], as well as an adaptation of an argument for the scalar case due to [Healy 08]. Our new multi-matrix Golden-Thompson inequality could be of independent interest. Secondarily, we also provide a generic reduction showing that any concentration inequality for vector-valued martingales implies a concentration inequality for the corresponding expander walk, with a weakening of parameters proportional to the squared mixing time.

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References

[1]

Rudolf Ahlswede and Andreas Winter. 2002. Strong converse for identification via quantum channels. IEEE Transactions on Information Theory 48, 3 (2002), 569–579.

Digital Library

[2]

Rajendra Bhatia. 1997.

[3]

Matrix analysis. Vol. 169. Springer Science & Business Media.

[4]

Herman Chernoff. 1952.

[5]

A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. The Annals of Mathematical Statistics (1952), 493–507.

[6]

Fan Chung and Linyuan Lu. 2006. Concentration inequalities and martingale inequalities: a survey. Internet Mathematics 3, 1 (2006), 79–127.

[7]

Kai-min Chung, Henry Lam, Zhenming Liu, and Michael Mitzenmacher. 2012.

[8]

Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified. arXiv preprint arXiv:1201.0559 (2012).

[9]

John B. Conway. 1978.

[10]

Functions of one Complex Variable. Graduate Texts in Mathematics (Second Edition), Vol. I. Springer New York, Heidelberg, Berlin.

[11]

David Gillman. 1998. A Chernoff bound for random walks on expander graphs. SIAM J. Comput. 27, 4 (1998), 1203–1220.

Digital Library

[12]

Sidney Golden. 1965. Lower bounds for the Helmholtz function. Physical Review 137, 4B (1965), B1127.

[13]

Loukas Grafakos. 2008.

[14]

Classical Fourier Analysis. Graduate Texts in Mathematics, Vol. 249. Springer New York.

[15]

Alexander D Healy. 2008. Randomness-efficient sampling within NC1. Computational Complexity 17, 1 (2008), 3–37.

Digital Library

[16]

Fumio Hiai, Robert Koenig, and Marco Tomamichel. 2016.

[17]

Generalized Log-Majorization and Multivariate Trace Inequalities. arXiv preprint arXiv:1609.01999 (Sept. 2016). arXiv: math-ph/1609.01999

[18]

Nabil Kahale. 1997. Large deviation bounds for Markov chains. Combinatorics, Probability and Computing 6, 04 (1997), 465–474.

Digital Library

[19]

Michel Ledoux and Michel Talagrand. 2013.

[20]

Probability in Banach Spaces: isoperimetry and processes. Springer Science & Business Media.

[21]

C. A. Leon and F. Perron. 2004. Optimal Hoeffding bounds for discrete reversible Markov chains. Annals of Applied Probability 14, 2 (2004).

[22]

Pascal Lezaud. 1998. Chernoff-type bound for finite Markov chains. Annals of Applied Probability (1998), 849–867.

[23]

Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. 1988. Ramanujan graphs. Combinatorica 8, 3 (1988), 261–277.

[24]

Assaf Naor, Yuval Peres, Oded Schramm, Scott Sheffield, et al. 2006.

[25]

Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Mathematical Journal 134, 1 (2006), 165–197.

[26]

Shravas Rao and Oded Regev. 2017. A Sharp Tail Bound for the Expander Random Sampler. arXiv preprint arXiv:1703.10205 (2017).

[27]

Omer Reingold, Salil Vadhan, and Avi Wigderson. 2000.

[28]

Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on. IEEE, 3–13.

Digital Library

[29]

Mark Rudelson. 1999.

[30]

Random vectors in the isotropic position. Journal of Functional Analysis 164, 1 (1999), 60–72.

[31]

David Sutter, Mario Berta, and Marco Tomamichel. 2017.

[32]

Multivariate Trace Inequalities. Communications in Mathematical Physics 352, 1 (2017), 37–58. 016- 2778- 5 arXiv:1604.03023.

[33]

Colin J Thompson. 1965. Inequality with applications in statistical mechanics. J. Math. Phys. 6, 11 (1965), 1812–1813.

[34]

Joel A Tropp. 2012.

[35]

User-friendly tail bounds for sums of random matrices. Foundations of computational mathematics 12, 4 (2012), 389–434.

[36]

Joel A. Tropp. 2015. An introduction to matrix concentration inequalities. Foundations and Trends® in Machine Learning 8, 1-2 (2015), 1–230.

Digital Library

[37]

R. Wagner. 2008. Tail estimates for sums of variables sampled by a random walk. Combinatorics, Probability and Computing 17, 2 (2008).

Digital Library

[38]

Avi Wigderson and David Xiao. 2005. A randomness-efficient sampler for matrixvalued functions and applications. In Foundations of Computer Science, 2005. FOCS 2005. 46th Annual IEEE Symposium on. IEEE, 397–406.

Digital Library

[39]

Avi Wigderson and David Xiao. 2006.

[40]

Retraction of Wigderson-Xiao: A randomness-efficient sampler for matrix valued functions, eccc tr05-107. (2006).

Cited By

Djouzi KBeghdad-Bey KAmamra A(2022)A Scalable Adaptive Sampling Based Approach for Big Data ClassificationAdvances in Computing Systems and Applications10.1007/978-3-031-12097-8_7(73-83)Online publication date: 28-Sep-2022
https://doi.org/10.1007/978-3-031-12097-8_7
Huang DTropp J(2021)From Poincaré inequalities to nonlinear matrix concentrationBernoulli10.3150/20-BEJ128927:3Online publication date: 1-May-2021
https://doi.org/10.3150/20-BEJ1289
Qiu JWang CLiao BPeng RTang JLarochelle HRanzato MHadsell RBalcan MLin H(2020)A matrix chernoff bound for Markov chains and its application to co-occurrence matricesProceedings of the 34th International Conference on Neural Information Processing Systems10.5555/3495724.3497271(18421-18432)Online publication date: 6-Dec-2020
https://dl.acm.org/doi/10.5555/3495724.3497271
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Index Terms

A matrix expander Chernoff bound
1. Theory of computation
  1. Models of computation
    1. Probabilistic computation
  2. Randomness, geometry and discrete structures

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cover image ACM Conferences

STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

June 2018

1332 pages

ISBN:9781450355599

DOI:10.1145/3188745

General Chairs:
Ilias Diakonikolas
University of Southern California, USA
,
David Kempe
University of Southern California, USA
,
Program Chair:
Monika Henzinger
University of Vienna, Austria

Copyright © 2018 ACM.

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Published: 20 June 2018

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STOC '18: Symposium on Theory of Computing

June 25 - 29, 2018

CA, Los Angeles, USA

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Cited By

Djouzi KBeghdad-Bey KAmamra A(2022)A Scalable Adaptive Sampling Based Approach for Big Data ClassificationAdvances in Computing Systems and Applications10.1007/978-3-031-12097-8_7(73-83)Online publication date: 28-Sep-2022
https://doi.org/10.1007/978-3-031-12097-8_7
Huang DTropp J(2021)From Poincaré inequalities to nonlinear matrix concentrationBernoulli10.3150/20-BEJ128927:3Online publication date: 1-May-2021
https://doi.org/10.3150/20-BEJ1289
Qiu JWang CLiao BPeng RTang JLarochelle HRanzato MHadsell RBalcan MLin H(2020)A matrix chernoff bound for Markov chains and its application to co-occurrence matricesProceedings of the 34th International Conference on Neural Information Processing Systems10.5555/3495724.3497271(18421-18432)Online publication date: 6-Dec-2020
https://dl.acm.org/doi/10.5555/3495724.3497271
Naor ARao SRegev O(2020)Concentration of Markov chains with bounded momentsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques10.1214/19-AIHP103956:3Online publication date: 1-Aug-2020
https://doi.org/10.1214/19-AIHP1039
Bakshi AChepurko NJayaram R(2020)Testing Positive Semi-Definiteness via Random Submatrices2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00114(1191-1202)Online publication date: Nov-2020
https://doi.org/10.1109/FOCS46700.2020.00114
Rao S(2019)A Hoeffding inequality for Markov chainsElectronic Communications in Probability10.1214/19-ECP21924:noneOnline publication date: 1-Jan-2019
https://doi.org/10.1214/19-ECP219
De AMossel ENeeman J(2019)Junta Correlation is Testable2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2019.00090(1549-1563)Online publication date: Nov-2019
https://doi.org/10.1109/FOCS.2019.00090
Kyng RSong Z(2018)A Matrix Chernoff Bound for Strongly Rayleigh Distributions and Spectral Sparsifiers from a few Random Spanning Trees2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2018.00043(373-384)Online publication date: Oct-2018
https://doi.org/10.1109/FOCS.2018.00043

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