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The algebraic degree of the Wasserstein distance

Chiara Meroni, Bernhard Reinke, Kexin Wang

Abstract

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 the structure of a subpolytope of the Birkhoff polytope, invariant under a transformation induced by complex conjugation.

The algebraic degree of the Wasserstein distance

Abstract

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 the structure of a subpolytope of the Birkhoff polytope, invariant under a transformation induced by complex conjugation.
Paper Structure (9 sections, 11 theorems, 34 equations, 3 figures, 2 tables)

This paper contains 9 sections, 11 theorems, 34 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Finitely supported probability measures
  3. Invariant Birkhoff polytope
  4. Graph interpretation
  5. Realization of the vertices
  6. Algebraic degree formulae
  7. Algebraic degrees
  8. Wasserstein distance degree via elimination
  9. Acknowledgements.

Key Result

Lemma 2.4

Let $p, q \in \mathbb{C}[z]$ be polynomials of degree $d$ with roots $\alpha_1, \dots, \alpha_d$ and $\beta_1, \dots, \beta_d$ respectively. Suppose that $p$ and $q$ have no multiple roots. Then the following is a bijection:

Figures (3)

  • Figure 1: The roots of $p$ (blue dots) and the roots of $q$ (red diamonds), from \ref{['ex:intro']}. The segments joining them represent a pairing that realizes the minimum for the Wasserstein distance of $p,q$.
  • Figure 2: The invariant square $B^\iota_3$ from \ref{['ex:invBthree_graph']}. Its vertices correspond to bipartite graphs $\Gamma$. The purple squares represent real roots, the green triangles represent complex roots. The involution $\iota$ exchanges pairs of green triangles.
  • Figure 3: Disjoint cycles in $\Gamma$ as subgraphs of the Eisenstein lattice. From left to right, we have disjoint cycles of types 1, 2 (pair), 4, 4, 3, 4.

Theorems & Definitions (36)

  • Example 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 26 more