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Degrees of the Wasserstein Distance to Small Toric Models

Greg DePaul, Serkan Hoşten, Nilava Metya, Ikenna Nometa

TL;DR

This work studies the algebraic complexity of computing Wasserstein distances from discrete empirical distributions to small toric models. It introduces the Wasserstein degree $w(X,F)$, the number of complex critical points of a linear objective on a translated face intersected with a toric model, and shows $w(X,F)$ is bounded by the polar degrees of the underlying variety. The authors derive explicit polar-degree formulas for rational normal scrolls and a range of graphical models (star trees, the binary 4-path, the binary 4-cycle, and no-three-way interaction models) and present an algorithm to compute Wasserstein degrees without Lagrange multipliers. Computational experiments, implemented in SageMath and Macaulay2, demonstrate that $w(X,F)$ is often smaller than the corresponding polar degrees, offering a practical handle on the complexity of Wasserstein-distance computations in discrete settings and informing distance-based estimation and learning tasks.

Abstract

The study of the closest point(s) on a statistical model from a given distribution in the probability simplex with respect to a fixed Wasserstein metric gives rise to a polyhedral norm distance optimization problem. There are two components to the complexity of determining the Wasserstein distance from a data point to a model. One is the combinatorial complexity that is governed by the combinatorics of the Lipschitz polytope of the finite metric to be used. Another is the algebraic complexity, which is governed by the polar degrees of the Zariski closure of the model. We find formulas for the polar degrees of rational normal scrolls and graphical models whose underlying graphs are star trees. Also, the polar degrees of the graphical models with four binary random variables where the graphs are a path on four vertices and the four-cycle, as well as for small, no-three-way interaction models, were computed. We investigate the algebraic degree of computing the Wasserstein distance to a small subset of these models. It was observed that this algebraic degree is typically smaller than the corresponding polar degree.

Degrees of the Wasserstein Distance to Small Toric Models

TL;DR

This work studies the algebraic complexity of computing Wasserstein distances from discrete empirical distributions to small toric models. It introduces the Wasserstein degree , the number of complex critical points of a linear objective on a translated face intersected with a toric model, and shows is bounded by the polar degrees of the underlying variety. The authors derive explicit polar-degree formulas for rational normal scrolls and a range of graphical models (star trees, the binary 4-path, the binary 4-cycle, and no-three-way interaction models) and present an algorithm to compute Wasserstein degrees without Lagrange multipliers. Computational experiments, implemented in SageMath and Macaulay2, demonstrate that is often smaller than the corresponding polar degrees, offering a practical handle on the complexity of Wasserstein-distance computations in discrete settings and informing distance-based estimation and learning tasks.

Abstract

The study of the closest point(s) on a statistical model from a given distribution in the probability simplex with respect to a fixed Wasserstein metric gives rise to a polyhedral norm distance optimization problem. There are two components to the complexity of determining the Wasserstein distance from a data point to a model. One is the combinatorial complexity that is governed by the combinatorics of the Lipschitz polytope of the finite metric to be used. Another is the algebraic complexity, which is governed by the polar degrees of the Zariski closure of the model. We find formulas for the polar degrees of rational normal scrolls and graphical models whose underlying graphs are star trees. Also, the polar degrees of the graphical models with four binary random variables where the graphs are a path on four vertices and the four-cycle, as well as for small, no-three-way interaction models, were computed. We investigate the algebraic degree of computing the Wasserstein distance to a small subset of these models. It was observed that this algebraic degree is typically smaller than the corresponding polar degree.
Paper Structure (11 sections, 9 theorems, 45 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 11 sections, 9 theorems, 45 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Polar Degrees
  3. Polar Degrees
  4. Rational Normal Scrolls
  5. Small graphical models
  6. Polar degrees of star trees
  7. Polar degrees of the binary $4$-path
  8. Polar degrees of the binary $4$-cycle
  9. Polar degrees of the no-three-way interaction models
  10. Computing Wasserstein Degrees
  11. Computational experiments

Key Result

Theorem 1.3

ccelik2021wasserstein The Wasserstein degree $w(X,F)$ where $\mathop{\mathrm{codim}}\nolimits(F)=r+1$ is at most $\mu_i(X)$, the $i$th polar degree of $X$ where $i=\dim(X)-r$.

Figures (6)

  • Figure 1: Wasserstein ball for the discrete metric touching the twisted cubic at an edge.
  • Figure 2: Polytope for the Hirzebruch surface $S(a,b)$.
  • Figure 3: A star tree with $n$ leaves.
  • Figure 4: Small path graphs on $2,3,$ and $4$ vertices.
  • Figure 5: The smallest nondecomposable graph (the $4-cycle$ graph).
  • ...and 1 more figures

Theorems & Definitions (24)

  • Example 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1: Polar variety BKSmetricAlgGeo
  • Definition 2.2: Polar degree
  • Proposition 2.3
  • Example 2.4: Rational Normal Quartic Curve
  • Definition 2.5: Conormal Variety
  • Theorem 2.6
  • Theorem 2.7
  • ...and 14 more