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A Constructive Method to Minimize the Index of Coincidence under Marginal Constraints

Pierre Jean-Claude Robert Bertrand

TL;DR

The paper addresses minimizing the index of coincidence $IC(\pi)=\sum_{u,v}\pi_{u,v}^2$ over couplings with fixed marginals $\mu,\nu$, providing a complete constructive solution. It shows a closed-form $\pi^* = \pi^+$ under Condition $H$, but also develops a general method yielding $\pi^*$ via a monotone staircase-zero structure; under a rectangle-zero assumption this reduces to an explicit transform $\tilde{\pi}^+$, and an iterative algorithm guarantees convergence to the optimal coupling in at most $p-1$ steps without margin restrictions. The work also quantifies how often Condition $H$ holds, finding that the eligible margin set has vanishing measure as dimension grows, thereby justifying the need for the general constructive approach. Collectively, the results offer a rigorous, finite-step, fully constructive solution to index-of-coincidence minimization under marginal constraints with broad applicability in information theory and related fields.

Abstract

We consider the problem of minimizing the index of coincidence of a joint distribution under fixed marginal constraints. This objective is motivated by several applications in information theory, where the index of coincidence naturally arises. A closed-form solution is known when the marginals satisfy a strong feasibility condition, but this condition is rarely met in practice. We first show that the measure of the set of marginals for which condition applies vanishes as the dimension grows. We then characterize the structure of the optimal coupling in the general case, proving that it exhibits a monotone staircase of zero entries. Based on this structure, we propose an explicit iterative construction and prove that it converges in finitely many steps to a minimizer. Main result of the paper is a complete constructive solution of index-of-coincidence minimization.

A Constructive Method to Minimize the Index of Coincidence under Marginal Constraints

TL;DR

The paper addresses minimizing the index of coincidence over couplings with fixed marginals , providing a complete constructive solution. It shows a closed-form under Condition , but also develops a general method yielding via a monotone staircase-zero structure; under a rectangle-zero assumption this reduces to an explicit transform , and an iterative algorithm guarantees convergence to the optimal coupling in at most steps without margin restrictions. The work also quantifies how often Condition holds, finding that the eligible margin set has vanishing measure as dimension grows, thereby justifying the need for the general constructive approach. Collectively, the results offer a rigorous, finite-step, fully constructive solution to index-of-coincidence minimization under marginal constraints with broad applicability in information theory and related fields.

Abstract

We consider the problem of minimizing the index of coincidence of a joint distribution under fixed marginal constraints. This objective is motivated by several applications in information theory, where the index of coincidence naturally arises. A closed-form solution is known when the marginals satisfy a strong feasibility condition, but this condition is rarely met in practice. We first show that the measure of the set of marginals for which condition applies vanishes as the dimension grows. We then characterize the structure of the optimal coupling in the general case, proving that it exhibits a monotone staircase of zero entries. Based on this structure, we propose an explicit iterative construction and prove that it converges in finitely many steps to a minimizer. Main result of the paper is a complete constructive solution of index-of-coincidence minimization.
Paper Structure (24 sections, 21 theorems, 106 equations, 2 figures, 2 algorithms)

This paper contains 24 sections, 21 theorems, 106 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Distribute the coupling across the matrix
  3. Motivations: minimizing index of coincidence
  4. Paper summary
  5. Form of the solution $\pi^*$
  6. The closed-form solution $\pi^* = \pi^+$, valid only when Condition \ref{['cond:H']} holds
  7. $\pi^+$ as a mixture of three independent couplings
  8. Deriving $\pi^*$ using the Lagrangian method
  9. A staircase structure containing the zeros
  10. Construction of the transformation $\pi^+\mapsto\tilde{\pi}^+$
  11. Assumption: a single rectangle of zeros
  12. Closed-form expression of $\pi^*=\tilde{\pi}^+$
  13. Understanding the structure of $\tilde{\pi}^+$
  14. Upper-left rectangle
  15. Upper-right rectangle
  16. ...and 9 more sections

Key Result

Proposition 1

The problem pb:chi2 admits a solution, which we denote by $\pi^*$.

Figures (2)

  • Figure 1: General shape of $\pi^*$: each row begins with a sequence of zeros, followed by strictly positive entries represented by $+$; the number of zeros decreases across rows.
  • Figure 2: Shape of $\pi^*$ under our assumption: the first $p_1$ rows contain $q_1$ zeros, while the remaining rows contain none

Theorems & Definitions (58)

  • Definition 1: Independence
  • Definition 2: Indeterminacy
  • Definition 3: Uniform law
  • Proposition 1: Existence of a solution
  • proof
  • Proposition 2
  • proof
  • Proposition 3: $\pi^+$ as a solution
  • proof
  • Remark 1
  • ...and 48 more