Contingency tables with uniformly bounded entries

TL;DR

This paper develops a concave approximation for counting nonnegative integer matrices with fixed margins and bounded entries, providing efficient estimators and revealing exponential growth in certain constrained matrix counts.

Contribution

It introduces a concave approximation for the logarithm of the count of such matrices and provides asymptotically exact estimators based on maximum-entropy models.

Findings

The logarithm of the count is approximated by a concave function.

Efficient estimators for the count are proposed and shown to be asymptotically exact.

The number of matrices exceeds heuristic predictions by an exponential factor under certain conditions.

Abstract

We consider nonnegative integer matrices with specified row and column sums and upper bounds on the entries. We show that the logarithm of the number of such matrices is approximated by a concave function of the row and column sums. We give efficiently computable estimators for this function, including one suggested by a maximum-entropy random model; we show that these estimators are asymptotically exact as the dimension of the matrices goes to infinity. We finish by showing that, for kappa >= 2 and for sufficiently small row and column sums, the number of matrices with these row and column sums and with entries <= kappa is greater by an exponential factor than predicted by a heuristic of independence.