Document

# Chernoff Bound for High-Dimensional Expanders

## File

LIPIcs.APPROX-RANDOM.2020.25.pdf
• Filesize: 0.53 MB
• 22 pages

## Cite As

Tali Kaufman and Ella Sharakanski. Chernoff Bound for High-Dimensional Expanders. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.25

## Abstract

We generalize the expander Chernoff bound to high-dimensional expanders. The expander Chernoff bound is an essential property of expanders, first proved by Gillman [Gillman, 1993]. Given a graph G and a function f on the vertices, it states that the probability of f’s mean sampled via a random walk on G to deviate from its actual mean, has a bound that depends on the spectral gap of the walk and decreases exponentially as the walk’s length increases. We are interested in obtaining an analog Chernoff bound for high order walks on high-dimensional expanders. A naive generalization of the expander Chernoff bound from expander graphs to high-dimensional expanders gives a very poor bound due to obstructions that occur in high-dimensional expanders and are not present in (one-dimensional) expander graphs. Because of these obstructions, the spectral gap of high-order random walks is inherently small. A natural question that arises is how to get a meaningful Chernoff bound for high-dimensional expanders. In this paper, we manage to get a strong Chernoff bound for high-dimensional expanders by looking beyond the spectral gap. First, we prove an expander Chernoff bound that depends on a notion that we call the "shrinkage of a function" instead of the spectral gap. In one-dimensional expanders, the shrinkage of any function with zero-mean is bounded by λ(M). Therefore, the spectral gap is just the one-dimensional manifestation of the shrinkage. Next, we show that in good high-dimensional expanders, the shrinkage of functions that "do not come from below" is good. A function does not come from below if from any local point of view (called "link") its mean is zero. Finally, we prove a high-dimensional Chernoff bound that captures the expansion of the complex. When the function on the faces has a small variance and does not "come from below", our bound is better than the naive high-dimensional expander Chernoff bound.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Random walks and Markov chains
• Mathematics of computing → Spectra of graphs
• Mathematics of computing → Hypergraphs
##### Keywords
• High Dimensional Expanders
• Random Walks
• Tail Bounds

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 1-12. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316385.
2. Sheldon Jay Axler. Linear algebra done right, volume 2. Springer, third edition, 1997.
3. Kazuoki Azuma. Weighted sums of certain dependent random variables. Tohoku Math. J. (2), 19(3):357-367, 1967. URL: https://doi.org/10.2748/tmj/1178243286.
4. Fan Chung and Linyuan Lu. Concentration inequalities and martingale inequalities: a survey. Internet Mathematics, 3(1):79-127, 2006.
5. Kai-Min Chung, Henry Lam, Zhenming Liu, and Michael Mitzenmacher. Chernoff-Hoeffding bounds for markov chains: Generalized and simplified. In 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, Paris, France, pages 124-135, 2012. URL: https://doi.org/10.4230/LIPIcs.STACS.2012.124.
6. Irit Dinur, Prahladh Harsha, Tali Kaufman, Inbal Livni Navon, and Amnon Ta Shma. List decoding with double samplers. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2134-2153. SIAM, 2019.
7. Irit Dinur and Tali Kaufman. High dimensional expanders imply agreement expanders. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 974-985, 2017. URL: https://doi.org/10.1109/FOCS.2017.94.
8. Ankit Garg, Yin Tat Lee, Zhao Song, and Nikhil Srivastava. A matrix expander chernoff bound. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 1102-1114, New York, NY, USA, 2018. ACM. URL: https://doi.org/10.1145/3188745.3188890.
9. D Gillman. A chernoff bound for random walks on expander graphs. In Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science, pages 680-691. IEEE, 1993.
10. David Gillman. A chernoff bound for random walks on expander graphs. SIAM Journal on Computing, 27(4):1203-1220, 1998.
11. Geoffrey Grimmett and David Stirzaker. Probability and Random Processes, pages 225, 227. Oxford University Press, third edition, 2001.
12. Alexander D Healy. Randomness-efficient sampling within nc. Computational Complexity, 17(1):3-37, 2008.
13. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bulletin of the American Mathematical Society, 43(4):439-561, 2006.
14. Nabil Kahale. Large deviation bounds for markov chains. Combinatorics, Probability and Computing, 6(4):465-474, 1997.
15. Tali Kaufman and David Mass. High dimensional random walks and colorful expansion. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017.
16. Tali Kaufman and Izhar Oppenheim. High Order Random Walks: Beyond Spectral Gap. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018), volume 116 of Leibniz International Proceedings in Informatics (LIPIcs), pages 47:1-47:17, 2018.
17. Tali Kaufman and Izhar Oppenheim. High order random walks: Beyond spectral gap. Combinatorica, pages 1-37, 2020.
18. Pascal Lezaud. Chernoff-type bound for finite markov chains. Annals of Applied Probability, pages 849-867, 1998.
19. Alexander Lubotzky. Expander graphs in pure and applied mathematics. Bulletin of the American Mathematical Society, 49(1):113-162, 2012.
20. Roy Wagner. Tail estimates for sums of variables sampled by a random walk. Combinatorics, Probability and Computing, 17(2):307-316, 2008.
21. Avi Wigderson and David Xiao. A randomness-efficient sampler for matrix-valued functions and applications. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pages 397-406. IEEE, 2005.