arXiv 1603.017992016-03-06math.PRcs.CC

半空间中噪声的稳定性和相关性

Noise Stability and Correlation with Half Spaces

Elchanan Mossel, Joe Neeman

备注：: 23 pages

摘要：

Benjamini，Kalai和Schramm证明了单调函数$f : \{-1,1\}^n \to \{-1,1\}$是噪声稳定的当且仅当它与半空间（一个形式为$\{x: \langle x, a\rangle \le b\}$的集合）相关。我们研究噪声稳定性的一般（不一定单调）函数的半空间的相关性。我们表明，一个功能$f: \{-1, 1\}^n \to \{-1, 1\}$是噪声稳定的当且仅当它成为相关的半空间时，我们修改$f$随机限制其坐标的一个常数部分。考虑随机限制是必要的：我们构造噪声稳定函数，其与任何半空间的相关性是$o(1)$。这些例子进一步满足了不同的约束与不同的半空间相关：对于任何固定的半空间，随机约束与它相关的概率为零。我们还提供了上述陈述的定量版本，以及适用于$\mathbb{R}^n$而不是离散立方体上的高斯测度的版本。我们的工作是由学习理论中的问题和最近的Khot和Moshkovitz的问题。

英文摘要：

Benjamini, Kalai and Schramm showed that a monotone function $f : \{-1,1\}^n \to \{-1,1\}$ is noise stable if and only if it is correlated with a half-space (a set of the form $\{x: \langle x, a\rangle \le b\}$). We study noise stability in terms of correlation with half-spaces for general (not necessarily monotone) functions. We show that a function $f: \{-1, 1\}^n \to \{-1, 1\}$ is noise stable if and only if it becomes correlated with a half-space when we modify $f$ by randomly restricting a constant fraction of its coordinates. Looking at random restrictions is necessary: we construct noise stable functions whose correlation with any half-space is $o(1)$. The examples further satisfy that different restrictions are correlated with different half-spaces: for any fixed half-space, the probability that a random restriction is correlated with it goes to zero. We also provide quantitative versions of the above statements, and versions that apply for the Gaussian measure on $\mathbb{R}^n$ instead of the discrete cube. Our work is motivated by questions in learning theory and a recent question of Khot and Moshkovitz.

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。

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