Bernoulli Correlations and Cut Polytopes

TL;DR

This paper characterizes the set of correlation vectors for symmetric Bernoulli variables as a polytope, establishes its connection to cut polytopes, and provides methods for correlation matrix realizability and sampling.

Contribution

It identifies the Bernoulli correlation polytope as a cut polytope and derives explicit isomorphisms, enabling linear programming approaches for correlation feasibility and sampling.

Findings

The correlation vectors form a polytope with vertices linked to uniform distributions on cube diagonals.

The correlation polytope is affinely isomorphic to the cut polytope, with an explicit transformation.

Linear programming can determine the realizability of a given correlation matrix.

Abstract

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.