Maximum entropy Edgeworth estimates of the number of integer points in polytopes

TL;DR

This paper develops an Edgeworth-corrected maximum entropy method to accurately estimate the number of integer solutions in polytopes, with applications to contingency tables and graphs with specified degree sequences.

Contribution

It introduces a novel Edgeworth correction to the maximum entropy approach for counting integer points in polytopes, improving accuracy for large dimensions.

Findings

The method provides asymptotically valid estimates for large-scale problems.

Demonstrated effectiveness on contingency tables with increasing size.

Validated approach for counting graphs with given degree sequences.

Abstract

Abstract: The number of points $x=(x_1 ,x_2 ,...x_n)$ that lie in an integer cube $C$ in $R^n$ and satisfy the constraints $\sum_j h_{ij}(x_j )=s_i ,1\le i\le d$ is approximated by an Edgeworth-corrected Gaussian formula based on the maximum entropy density $p$ on $x \in C$, that satisfies $E\sum_j h_{ij}(x_j )=s_i ,1\le i\le d$. Under $p$, the variables $X_1 ,X_2 ,...X_n $ are independent with densities of exponential form. Letting $S_i$ denote the random variable $\sum_j h_{ij}(X_j )$, conditional on $S=s, X$ is uniformly distributed over the integers in $C$ that satisfy $S=s$. The number of points in $C$ satisfying $S=s$ is $p \{S=s\}\exp (I(p))$ where $I(p)$ is the entropy of the density $p$. We estimate $p \{S=s\}$ by $p_Z(s)$, the density at $s$ of the multivariate Gaussian $Z$ with the same first two moments as $S$; and when $d$ is large we use in addition an Edgeworth factor that requires the first four moments of $S$ under $p$. The asymptotic validity of the Edgeworth-corrected estimate is proved and demonstrated for counting contingency tables with given row and column sums as the number of rows and columns approaches infinity, and demonstrated for counting the number of graphs with a given degree sequence, as the number of vertices approaches infinity.