Bernoulli相关与Cut多面体
Bernoulli Correlations and Cut Polytopes
摘要：
给定$n$对称伯努利变量，可以说它们的相关阵是一个向量吗？我们证明了这些向量的集合$R(\mathcal{B}_n)$是一个多面体，并确定其顶点。这些极值点对应于与立方体$[0,1]^n$对角线上的离散均匀分布相关联的相关向量。我们还表明，多面体是仿射同构的一个著名的切割多面体${\rm CUT}(n)$被定义为一个凸壳的切割向量在一个完全图的顶点集$\{1,\ldots,n\}$。同构被明确地获得为$R(\mathcal{B}_n)= {\mathbf{1}}-2~{\rm CUT}(n)$。作为这项工作的必然结果，它是直接使用线性规划，以确定是否一个特定的相关阵是可实现的或没有。进而得到了具有给定相关性的多元对称Bernoullis的抽样方法。在某些情况下，该方法也可以用于一般的，而不仅仅是伯努利，边缘。
英文摘要：
Given $n$ symmetric Bernoulli variables, what can be said about their correlation matrix viewed as a vector? We show that the set of those vectors $R(\mathcal{B}_n)$ is a polytope and identify its vertices. Those extreme points correspond to correlation vectors associated to the discrete uniform distributions on diagonals of the cube $[0,1]^n$. We also show that the polytope is affinely isomorphic to a well-known cut polytope ${\rm CUT}(n)$ which is defined as a convex hull of the cut vectors in a complete graph with vertex set $\{1,\ldots,n\}$. The isomorphism is obtained explicitly as $R(\mathcal{B}_n)= {\mathbf{1}}-2~{\rm CUT}(n)$. As a corollary of this work, it is straightforward using linear programming to determine if a particular correlation matrix is realizable or not. Furthermore, a sampling method for multivariate symmetric Bernoullis with given correlation is obtained. In some cases the method can also be used for general, not exclusively Bernoulli, marginals.
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