Log concavity and concentration of Lipschitz functions on the Boolean hypercube

Abstract: It is well-known that measures whose density is the form $e^{-V}$ where $V$ is a uniformly convex potential on $\RR^n$ attain strong concentration properties. In search of a notion of log-concavity on the discrete hypercube, we consider measures on ${-1,1}^n$ whose multi-linear extension $f$ satisfies $\log \nabla^2 f(x) \preceq β\Id$, for $β\geq 0$, which we refer to as $β$-semi-log-concave. We prove that these measures satisfy a nontrivial concentration bound, namely, any Hamming Lipchitz test function $\varphi$ satisfies $\Var_ν[\varphi] \leq n^{2-C_β}$ for $C_β>0$. As a corollary, we prove a concentration bound for measures which exhibit the so-called Rayleigh property. Namely, we show that for measures such that under any external field (or exponential tilt), the correlation between any two coordinates is non-positive, Hamming-Lipschitz functions admit nontrivial concentration.

• The paper introduces β-semi-log-concavity for measures on the Boolean hypercube, providing new concentration inequalities for Lipschitz functions.
• It demonstrates that negatively dependent Bernoulli sequences with the Rayleigh property yield nontrivial variance bounds for Hamming-Lipschitz functions.
• The study establishes entropy bounds linking discrete log-concavity to product measure approximations and outlines potential algorithmic applications.

Log Concavity and Concentration of Lipschitz Functions on the Boolean Hypercube

Introduction

The paper "Log concavity and concentration of Lipschitz functions on the Boolean hypercube" (2007.13108) explores the concentration properties of measures on the Boolean hypercube, $\{-1,1\}^n$. In continuous spaces, measures with densities of the form $e^{-V}$, where $V$ is a convex potential, possess strong concentration properties. This work seeks to extend the notion of log-concavity into discrete spaces, which are characterized by the Boolean hypercube, and establish concentration inequalities analogous to those found in Euclidean spaces.

Log-Concavity in the Discrete Setting

The crux of the research is the introduction of a new concept termed $\beta$-semi-log-concavity for measures on the Boolean hypercube. This definition requires the multi-linear extension of the measure, $f$, to satisfy $\log \nabla^2 f(x) \preceq \beta \Id$ for some $\beta \geq 0$. Measures meeting this criterion are shown to adhere to a concentration inequality: any Hamming-Lipschitz function $\varphi$ satisfies $\Var_\nu[\varphi] \leq n^{2-C_\beta}$ for a positive constant $C_\beta$.

Concentration of Negatively Dependent Random Variables

An intriguing implication of the research is its applicability to sequences of Bernoulli random variables exhibiting negative dependence. The paper introduces the Rayleigh property — a new class of negatively dependent measures. It is shown that for measures with the Rayleigh property, or where the correlation between any two coordinates becomes non-positive under an exponential tilt, Hamming-Lipschitz functions achieve nontrivial concentration bounds. Specifically, $\Var[\varphi(X_1,\dots,X_n)] \leq C n^{2-c}$ is guaranteed for universal constants $C$ and $c$.

Entropy Bound

In parallel with concentration results, the paper also provides bounds on entropy. It describes conditions under which a measure's entropy closely approximates that of a product measure with identical marginals. The entropy $\mathcal{H}(\nu)$ and a comparison term $\tilde {\mathcal{H}(\nu)}$, which sums the entropies of marginals, are related such that under a specific log-concavity-type condition, the inequality $$\tilde {\mathcal{H}(\nu)} \leq \beta \mathcal{H}(\nu)$$ holds.

Implications and Future Directions

The theoretical framework laid out in this paper has substantial implications for understanding concentration phenomena in discrete settings. It opens up new pathways for exploring concentration in structures not typically associated with log-concavity. Future research could leverage these insights in designing and analyzing algorithms within discrete domains, focusing on stochastic processes or mixing times in Markov models. Additionally, exploring stronger versions of log-concavity and further relations to polynomial-time solvability of certain combinatorial problems may prove fruitful.

Conclusion

Through rigorous analytical methods, the paper advances our understanding of concentration inequalities in Boolean spaces by introducing the concept of $\beta$-semi-log-concavity. The novel approach to handling negatively dependent variables and entropy bounds broadens the scope of log-concavity beyond continuous settings. As such, this research offers new mathematical tools that are potentially beneficial in theoretical exploration and practical applications within discrete settings.