布尔超立方体上Lipschitz函数的对数收敛与集中
Log concavity and concentration of Lipschitz functions on the Boolean hypercube
摘要：
众所周知，密度为$e^{-V}$形式的测度获得强的浓度性质，其中$V$是$\RR^n$上的一致凸势。为了寻找离散超立方体上的对数凹的概念，我们考虑$\{-1,1\}^n$上的测度，其多线性扩张$f$满足$\log \nabla^2 f(x) \preceq β\Id$，对于$β\geq 0$，我们称之为$β$ -半对数凹.我们证明了这些措施满足一个非平凡的浓度界，即任何Hamming Lipchitz测试功能$φ$满足$\Var_ν[φ] \leq n^{2-C_β}$的$C_β>0$。作为一个推论，我们证明了一个浓度界的措施，表现出所谓的瑞利性质。也就是说，我们表明，措施，使在任何外部领域（或指数倾斜），任何两个坐标之间的相关性是非正的，Hamming-Lipschitz功能允许非平凡的浓度。
英文摘要：
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 $φ$ satisfies $\Var_ν[φ] \leq n^{2-C_β}$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
