Chernoff Bounds and Reverse Hypercontractivity on HDX (2404.10961v2)

Abstract: We prove optimal concentration of measure for lifted functions on high dimensional expanders (HDX). Let $X$ be a $k$-dimensional HDX. We show for any $i\leq k$ and $f:X(i)\to [0,1]$: [\Pr_{s\in X(k)}\left[\left|\underset{{t\subseteq s}}{\mathbb{E}}[f(t)]-\mu\right|\geq\varepsilon\right]\leq exp\left(-\varepsilon^2\frac{k}{i}\right).] Using this fact, we prove that high dimensional expanders are reverse hypercontractive, a powerful functional inequality from discrete analysis implying that for any sets $A,B \subset X(k)$, the probability a $\rho$-correlated pair passes between them is at least [\Pr_{s,s' \sim T_\rho}[s \in A, s' \in B] \geq \Pr[A]^{O(1)} \Pr[B]^{O(1)}.] Our results hold under weak spectral assumptions on $X$. Namely we prove exponential concentration of measure for any complex below the Trickling-Down Threshold' (beyond which concentration may be arbitrarily poor), and optimal concentration for $\sqrt{k}$-skeletons of such complexes. We also show optimal bounds for the top dimension of stronger HDX among other settings. We leverage our inequalities to prove several new agreement testing theorems on high dimensional expanders, including a new 99%-regime test for subsets, and a variant of theZ-test' achieving inverse exponential soundness under the stronger assumption of $\ell_\infty$-expansion. The latter gives rise to the first optimal testers beyond the complete complex and products, a stepping stone toward the use of HDX in strong soundness PCPs. We also give applications within expansion, analysis, combinatorics, and coding theory, including a proof that two-sided HDX have optimal geometric overlap (giving the first explicit bounded-degree construction), near-optimal double samplers, new super-exponential degree lower bounds for certain HDX, distance-amplified list-decodable and locally testable codes, a Frankl-R\"odl Theorem and more.

• The paper establishes optimal concentration of lifted functions on HDX by proving tight Chernoff bounds and reverse hypercontractivity.
• It employs rigorous combinatorial and probabilistic techniques to derive agreement testing theorems with significant implications in computational complexity.
• The results pave the way for advanced applications in coding theory, aiding the construction of locally-testable codes and robust error-correcting schemes.

Chernoff Bounds and Reverse Hypercontractivity on HDX

This paper by Yotam Dikstein and Max Hopkins addresses the convergence of measure properties for lifted functions on high-dimensional expanders (HDX) and explores the implications of reverse hypercontractivity within these combinatorial structures. The work is significant in theoretical computer science and mathematics for studying properties of HDX, which are core tools in modern applications, including coding theory, quantum complexity, and PCPs.

Main Contributions

The paper's primary contribution is the demonstration of optimal concentration of measure for lifted functions on HDX, providing tight bounds on the deviation of functions, such as those completable via a Chernoff-Hoeffding approach. Specifically, they assert for any $i < k$ and function $f: \X[i] \to [0,1]$: $\mathbb{P}_{s \in \X[k]} \left[\left|\mathbb{E}_{t \subseteq s}[f(t)] - \mu \right| > \varepsilon \right] \leq \exp\left(-\varepsilon^2 \frac{k}{i}\right).$ This result implies that HDX are reverse hypercontractive and strongly supports their applicability in discrete analysis and functional inequalities.

Analytical Approach

The authors adopt a combinatorial and probabilistic framework, applying rigorous mathematical analysis to prove exponential concentration for HDX across various scenarios, including under weak spectral assumptions. This results in several new agreement testing theorems pertinent to computational complexity and coding theory. They also introduce powerful applications for HDX, notably in ensuring optimal geometric overlap and in constructing novel lists of locally testable codes.

Implications

1. Functional Inequalities: The paper’s core result regarding reverse hypercontractivity, a functional inequality from discrete analysis, marks a significant theoretical contribution. It indicates that for any $\rho \in (0,1)$, $k<d$, probabilities for $\rho$-correlated variables show concentration behavior over sets $A$ and $B$ with probability at least:

$$\mathbb{P}_{s, s'}[s \in A, s' \in B] \geq |A|^{O(1)} |B|^{O(1)}.$$

1. Computational Complexity: By deploying HDX, the research contributes to advancements in PCPs and hardness of approximation, crucial domains within computational complexity theory. The independence of each subgraph emphasizes HDX's utility in PCP constructions with low error rates and strong soundness, further aligning with ongoing research attempting to tackle the Sliding Scale Conjecture.
2. Applications in Coding Theory: The authors present implications in coding theory, namely high-dimensional expanders' roles in constructing large distance list-decodable and locally-testable codes, providing improved error-correcting capacities while maintaining practical constraint sizes for implementations.

Future Directions

Potential future explorations could involve leveraging the techniques in varied computational problems needing robustness under random perturbations or testing under agreement contexts. Additionally, refining the constructed bounds opens avenues towards unlocking further dimensions in approximation algorithms with high fidelity and reduced complexity.

In summary, this work innovatively applies theoretical approaches to reinforce understanding and capabilities of high-dimensional expanders. It presents a foundation that enhances the utility of HDX in complexity theory, establishing implications that extend computational theory's horizons into broader practical territory.