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Optimal Transport and Wasserstein Barycenter for Radially Contoured Distributions

Keyu Chen, Yunxin Zhang

TL;DR

This work addresses optimal transport and Wasserstein barycenters for radially contoured distributions, a relaxation of elliptically contoured families. It derives a closed-form Monge map between radially contoured laws with potentially different generator functions, computes the corresponding Wasserstein distance, and shows that the Wasserstein barycenter of radially contoured distributions remains radially contoured (generalized when generators differ). It also provides two numerical counterexamples demonstrating that the barycenter of elliptically contoured distributions need not be elliptically contoured. The results broaden OT theory for non-Gaussian families and inform mixture modeling with heterogeneous generator functions, highlighting both preserved structure in the radial setting and its limits in the elliptical case.

Abstract

The optimal transport and Wasserstein barycenter of Gaussian distributions have been solved. In literature, the closed form formulas of the Monge map, the Wasserstein distance and the Wasserstein barycenter have been given. Moreover, when Gaussian distributions extend more generally to elliptically contoured distributions, similar results also hold true. In this case, Gaussian distributions are regarded as elliptically contoured distribution with generator function $e^{-x/2}$. However, there are few results about optimal transport for elliptically contoured distributions with different generator functions. In this paper, we degenerate elliptically contoured distributions to radially contoured distributions and study their optimal transport and prove their Wasserstein barycenter is still radially contoured. For general elliptically contoured distributions, we give two numerical counterexamples to show that the Wasserstein barycenter of elliptically contoured distributions does not have to be elliptically contoured.

Optimal Transport and Wasserstein Barycenter for Radially Contoured Distributions

TL;DR

This work addresses optimal transport and Wasserstein barycenters for radially contoured distributions, a relaxation of elliptically contoured families. It derives a closed-form Monge map between radially contoured laws with potentially different generator functions, computes the corresponding Wasserstein distance, and shows that the Wasserstein barycenter of radially contoured distributions remains radially contoured (generalized when generators differ). It also provides two numerical counterexamples demonstrating that the barycenter of elliptically contoured distributions need not be elliptically contoured. The results broaden OT theory for non-Gaussian families and inform mixture modeling with heterogeneous generator functions, highlighting both preserved structure in the radial setting and its limits in the elliptical case.

Abstract

The optimal transport and Wasserstein barycenter of Gaussian distributions have been solved. In literature, the closed form formulas of the Monge map, the Wasserstein distance and the Wasserstein barycenter have been given. Moreover, when Gaussian distributions extend more generally to elliptically contoured distributions, similar results also hold true. In this case, Gaussian distributions are regarded as elliptically contoured distribution with generator function . However, there are few results about optimal transport for elliptically contoured distributions with different generator functions. In this paper, we degenerate elliptically contoured distributions to radially contoured distributions and study their optimal transport and prove their Wasserstein barycenter is still radially contoured. For general elliptically contoured distributions, we give two numerical counterexamples to show that the Wasserstein barycenter of elliptically contoured distributions does not have to be elliptically contoured.
Paper Structure (6 sections, 10 theorems, 64 equations, 1 figure)

This paper contains 6 sections, 10 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of related results
  3. Optimal transport for radially contoured distributions
  4. Wasserstein barycenter of radially contoured distributions
  5. Numerical counterexamples for elliptically contoured distributions
  6. Conclusion and discussion

Key Result

Theorem 2.1

Given two measures $\mu_0$ and $\mu_1$ on ${\mathbb R}^d$. Suppose $\mu_0$ has a density $\rho$ with respect to the Lebesgue measure, then the optimal $\gamma$ in $(W2)$ is unique and supported on the graph $(x,T(x))$ of a Monge map $T: {\mathbb R}^d \to {\mathbb R}^d$. It means $\gamma = (\operator

Figures (1)

  • Figure 1: The counterexamples that the Wasserstein barycenter of two elliptically contoured distributions may not be elliptically contoured. The first row is the first counterexample and the second row is the second one. In each row, the first two figures plot the contours of marginals, and the last figure shows the contours of the barycenter with weights $(0.5,0.5)$.

Theorems & Definitions (21)

  • Theorem 2.1: Brenier BrenierYann1991Pfam, PeyreGabriel2019Cot
  • Theorem 2.2: Characterization of barycenters AguehMartial2011Bitw
  • Remark 2.3
  • Definition 3.1: Radially contoured distribution
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Remark 3.6
  • Lemma 3.7
  • ...and 11 more