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oai:drops-oai.dagstuhl.de:12628 2024-03-06T10:51:17Z ddc:004 open_access

Chernoff Bound for High-Dimensional Expanders Kaufman, Tali Sharakanski, Ella High Dimensional Expanders Random Walks Tail Bounds 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Tali Kaufman and Ella Sharakanski 2020 Is Part Of LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2020.25 urn:nbn:de:0030-drops-126287 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.25 eng https://creativecommons.org/licenses/by/3.0/legalcode