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<p>Given two rational univariate polynomials, the Wasserstein distance of their associated measures is an algebraic number. We determine the algebraic degree of the squared Wasserstein distance, serving as a measure of algebraic complexity of the corresponding optimization problem. The computation relies on Galois theory and on the combinatorial structure of a specific subpolytope of the Birkhoff polytope, invariant under a transformation induced by complex conjugation. Proofs and computations also draw on notions from graph theory and elimination ideals.</p>

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The algebraic degree of the Wasserstein distance

  • 期刊论文

  • 发表日期：

    2025年6月2日

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  • Chiara Meroni,
  • Bernhard Reinke,
  • Kexin Wang

摘要

Given two rational univariate polynomials, the Wasserstein distance of their associated measures is an algebraic number. We determine the algebraic degree of the squared Wasserstein distance, serving as a measure of algebraic complexity of the corresponding optimization problem. The computation relies on Galois theory and on the combinatorial structure of a specific subpolytope of the Birkhoff polytope, invariant under a transformation induced by complex conjugation. Proofs and computations also draw on notions from graph theory and elimination ideals.