🤖 AI Summary
This work addresses a long-standing open problem in information theory: the lack of a general constructive solution for minimizing the index of coincidence of a joint distribution under given marginal constraints. The study provides the first complete characterization of the structural properties of optimal couplings, revealing that their zero entries exhibit a monotone staircase pattern. Building on this insight, the authors propose an explicit iterative construction algorithm that provably converges in finitely many steps to the global optimum for any feasible pair of marginals. The approach integrates tools from probabilistic coupling theory, combinatorial optimization, and iterative construction techniques, complemented by an asymptotic analysis of the measure of high-dimensional feasible sets, thereby resolving the constructive solvability of the index-of-coincidence minimization problem.
📝 Abstract
We study the problem of maximizing R{\'e}nyi entropy of order $2$ (equivalently, minimizing the index of coincidence) over the set of joint distributions with prescribed marginals. A closed-form optimizer is known under a feasibility condition on the marginals; we show that this condition is highly restrictive. We then provide an explicit construction of an optimal coupling for arbitrary marginals. Our approach characterizes the optimizer's structure and yields an iterative algorithm that terminates in finite time, returning an exact solution after at most $p-1$ updates, where $p$ is the number of rows.