← 研踪 PaperTrailarXiv:2601.09347 · v4
Maximizing R{é}nyi Entropy over the Transportation Polytope: Structure and Algorithm
首次提交 2026-01-14 18:22
版本更新 2026-10-28 16:34
雷达观测 2026-10-30 10:22
摘要
We study the problem of finding the minimum $\mathbb{L}\_2$-norm element in a transportation polytope. It corresponds to minimizing the index of coincidence or maximizing R{é}nyi entropy over joint probability distributions with prescribed marginals. The solution is straightforward under a feasibility condition on the marginals. We make three contributions. First, we decompose the basic solution into a mixture of three interpretable couplings and derive a closed-form sampling procedure, revealing its role as the $\mathbb{L}\_2$-projection of the uniform distribution onto the transportation polytope. Second, we show that the feasibility condition becomes exponentially rare as the number of modalities grows, which highlights the practical necessity of a more general construction. Third, we provide a KKT-based iterative construction of the optimal coupling for arbitrary marginals with explicit Lagrange multipliers.
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v4 · 2026-10-28 16:34 · 最新观测版本
Maximizing R{é}nyi Entropy over the Transportation Polytope: Structure and Algorithm
We study the problem of finding the minimum $\mathbb{L}\_2$-norm element in a transportation polytope. It corresponds to minimizing the index of coincidence or maximizing R{é}nyi entropy over joint probability distributions with prescribed marginals. The solution is straightforward under a feasibility condition on the marginals. We make three contributions. First, we decompose the basic solution into a mixture of three interpretable couplings and derive a closed-form sampling procedure, revealing its role as the $\mathbb{L}\_2$-projection of the uniform distribution onto the transportation polytope. Second, we show that the feasibility condition becomes exponentially rare as the number of modalities grows, which highlights the practical necessity of a more general construction. Third, we provide a KKT-based iterative construction of the optimal coupling for arbitrary marginals with explicit Lagrange multipliers.
查看此版本解读 →查看此版本原文 ↗v3 · 2026-10-27 22:20
A Constructive Method to Maximize Entropy under Marginal Constraints
We study the problem of maximizing R{é}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.
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