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Constrained Hellinger-Kantorovich barycenters: least-cost soft and conic multi-marginal formulations

Maciej Buze

TL;DR

This work addresses computing barycenters under the Hellinger-Kantorovich ($HK$) metric in unbalanced settings by introducing a constrained least-cost soft multi-marginal formulation with a one-sided hard marginal constraint. It then develops a conic multi-marginal reformulation based on a single joint perspective cost, and connects this constrained framework to unbalanced multi-marginal optimal transport through the notion of least cost. The approach is demonstrated with Dirac masses and Gaussian measures, illustrating numerical viability and highlighting the natural extension of Wasserstein barycenters to the unbalanced setting. Overall, the paper provides a principled, computationally tractable framework for least-cost HK barycenters that extends classical barycenters to unbalanced measures with potential wide-ranging applications.

Abstract

We show that the problem of finding the barycenter in the Hellinger-Kantorovich setting admits a least-cost soft multi-marginal formulation, provided that a one-sided hard marginal constraint is introduced. The constrained approach is then shown to admit a conic multi-marginal reformulation based on defining a single joint multi-marginal perspective cost function in the conic multi-marginal formulation, as opposed to separate two-marginal perspective cost functions for each two-marginal problem in the coupled-two-marginal formulation, as was studied previously in literature. We further establish that, as in the Wasserstein metric, the recently introduced framework of unbalanced multi-marginal optimal transport can be reformulated using the notion of the least cost. Subsequently, we discuss an example when input measures are Dirac masses and numerically solve an example for Gaussian measures. Finally, we also explore why the constrained approach can be seen as a natural extension of a Wasserstein space barycenter to the unbalanced setting.

Constrained Hellinger-Kantorovich barycenters: least-cost soft and conic multi-marginal formulations

TL;DR

This work addresses computing barycenters under the Hellinger-Kantorovich () metric in unbalanced settings by introducing a constrained least-cost soft multi-marginal formulation with a one-sided hard marginal constraint. It then develops a conic multi-marginal reformulation based on a single joint perspective cost, and connects this constrained framework to unbalanced multi-marginal optimal transport through the notion of least cost. The approach is demonstrated with Dirac masses and Gaussian measures, illustrating numerical viability and highlighting the natural extension of Wasserstein barycenters to the unbalanced setting. Overall, the paper provides a principled, computationally tractable framework for least-cost HK barycenters that extends classical barycenters to unbalanced measures with potential wide-ranging applications.

Abstract

We show that the problem of finding the barycenter in the Hellinger-Kantorovich setting admits a least-cost soft multi-marginal formulation, provided that a one-sided hard marginal constraint is introduced. The constrained approach is then shown to admit a conic multi-marginal reformulation based on defining a single joint multi-marginal perspective cost function in the conic multi-marginal formulation, as opposed to separate two-marginal perspective cost functions for each two-marginal problem in the coupled-two-marginal formulation, as was studied previously in literature. We further establish that, as in the Wasserstein metric, the recently introduced framework of unbalanced multi-marginal optimal transport can be reformulated using the notion of the least cost. Subsequently, we discuss an example when input measures are Dirac masses and numerically solve an example for Gaussian measures. Finally, we also explore why the constrained approach can be seen as a natural extension of a Wasserstein space barycenter to the unbalanced setting.
Paper Structure (29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 14 more sections

Key Result

Theorem 6.1

\newlabelthm:mvt0 Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure 1: Example figure using external image files.
  • Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

  • Theorem 6.1: Mean Value Theorem
  • Corollary 6.2
  • Proof 1
  • Claim 6.3
  • Proof 2: Proof of main theorem