On Approximability of Satisfiable k-CSPs: IV

Amey Bhangale, Subhash Khot, Dor Minzer
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引用次数: 3

Abstract

We prove a stability result for general $3$-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if $\Sigma,\Gamma$ and $\Phi$ are alphabets of constant size, and $\mu$ is a pairwise connected distribution over $\Sigma\times\Gamma\times\Phi$ with no $(\mathbb{Z},+)$ embeddings in which the probability of each atom is $\Omega(1)$, then the following holds. Any triplets of $1$-bounded functions $f\colon \Sigma^n\to\mathbb{C}$, $g\colon \Gamma^n\to\mathbb{C}$, $h\colon \Phi^n\to\mathbb{C}$ satisfying \[ \left|\mathbb{E}_{(x,y,z)\sim \mu^{\otimes n}}\big[f(x)g(y)h(z)\big]\right|\geq \varepsilon \] must arise from an Abelian group associated with the distribution $\mu$. More specifically, we show that there is an Abelian group $(H,+)$ of constant size such that for any such $f,g$ and $h$, the function $f$ (and similarly $g$ and $h$) is correlated with a function of the form $\tilde{f}(x) = \chi(\sigma(x_1),\ldots,\sigma(x_n)) L (x)$, where $\sigma\colon \Sigma \to H$ is some map, $\chi\in \hat{H}^{\otimes n}$ is a character, and $L\colon \Sigma^n\to\mathbb{C}$ is a low-degree function with bounded $2$-norm. En route we prove a few additional results that may be of independent interest, such as an improved direct product theorem, as well as a result we refer to as a ``restriction inverse theorem'' about the structure of functions that, under random restrictions, with noticeable probability have significant correlation with a product function. In companion papers, we show applications of our results to the fields of Probabilistically Checkable Proofs, as well as various areas in discrete mathematics such as extremal combinatorics and additive combinatorics.
可满足k- csp的近似性:IV
我们证明了在满足温和连通性的分布上的一般$3$明智相关性的稳定性结果。更具体地说,我们表明,如果$\Sigma,\Gamma$和$\Phi$是恒定大小的字母,并且$\mu$是$\Sigma\times\Gamma\times\Phi$上的成对连接分布,其中没有$(\mathbb{Z},+)$嵌入,其中每个原子的概率为$\Omega(1)$,则以下情况成立。满足\[ \left|\mathbb{E}_{(x,y,z)\sim \mu^{\otimes n}}\big[f(x)g(y)h(z)\big]\right|\geq \varepsilon \]的任何$1$ -有界函数$f\colon \Sigma^n\to\mathbb{C}$, $g\colon \Gamma^n\to\mathbb{C}$, $h\colon \Phi^n\to\mathbb{C}$的三元组必须产生于与分布$\mu$相关联的阿贝尔群。更具体地说,我们证明了存在一个大小不变的阿贝尔群$(H,+)$,使得对于任何这样的$f,g$和$h$,函数$f$(以及类似的$g$和$h$)与形式为$\tilde{f}(x) = \chi(\sigma(x_1),\ldots,\sigma(x_n)) L (x)$的函数相关,其中$\sigma\colon \Sigma \to H$是某个映射,$\chi\in \hat{H}^{\otimes n}$是一个字符,$L\colon \Sigma^n\to\mathbb{C}$是一个低度函数,具有有界的$2$范数。在此过程中,我们证明了一些可能独立感兴趣的其他结果,例如改进的直接乘积定理,以及我们称之为“限制逆定理”的结果,该结果是关于函数的结构,在随机限制下,与乘积函数具有显著的概率相关性。在同伴论文中,我们展示了我们的结果在概率可检验证明领域的应用,以及离散数学的各个领域,如极值组合学和加性组合学。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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