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Universal vector and matrix optimal transport

Boris Khesin, Klas Modin

TL;DR

This work develops a unified gauge-theoretic framework for optimal transport of vector and matrix densities by enlarging the symmetry group to semidirect products of diffeomorphisms with gauge transformations. It constructs Bures-type $L^2$ metrics and defines vector Wasserstein $WV$ and matrix Wasserstein $WM$ distances via Riemannian submersions, yielding explicit geodesic equations that generalize Burgers-type dynamics to both vector and matrix settings. The theory covers balanced and unbalanced transport, with detailed Hamiltonian formulations and reduced equations for the velocity field $\mathbf{u}$, gauge fields, and matrix/vector densities, connected through momentum maps and Poisson geometry. Applications span colored medical imaging, diffusion tensor imaging, quantum spin liquids, and multivariate time-series analysis, highlighting the practical impact of a universal transport framework for non-scalar densities. The approach provides a flexible platform for future extensions to bundles, other gauge groups, and relationships to generalized flows and Kähler structures.

Abstract

In this paper we propose a gauge-theoretic approach to the problems of optimal mass transport for vector and matrix densities. This resolves both the issues of positivity and action transitivity constraints. Bures-type metrics on the corresponding semi-direct product groups of diffeomorphisms and gauge transformations are related to Wasserstein-type metrics on vector half-densities and matrix densities via Riemannian submersions. We also describe their relation to Poisson geometry and demonstrate how the momentum map allows one to prove the Riemannian submersion properties. The obtained geodesic equations turn out to be vector versions of the Burgers equations.

Universal vector and matrix optimal transport

TL;DR

This work develops a unified gauge-theoretic framework for optimal transport of vector and matrix densities by enlarging the symmetry group to semidirect products of diffeomorphisms with gauge transformations. It constructs Bures-type metrics and defines vector Wasserstein and matrix Wasserstein distances via Riemannian submersions, yielding explicit geodesic equations that generalize Burgers-type dynamics to both vector and matrix settings. The theory covers balanced and unbalanced transport, with detailed Hamiltonian formulations and reduced equations for the velocity field , gauge fields, and matrix/vector densities, connected through momentum maps and Poisson geometry. Applications span colored medical imaging, diffusion tensor imaging, quantum spin liquids, and multivariate time-series analysis, highlighting the practical impact of a universal transport framework for non-scalar densities. The approach provides a flexible platform for future extensions to bundles, other gauge groups, and relationships to generalized flows and Kähler structures.

Abstract

In this paper we propose a gauge-theoretic approach to the problems of optimal mass transport for vector and matrix densities. This resolves both the issues of positivity and action transitivity constraints. Bures-type metrics on the corresponding semi-direct product groups of diffeomorphisms and gauge transformations are related to Wasserstein-type metrics on vector half-densities and matrix densities via Riemannian submersions. We also describe their relation to Poisson geometry and demonstrate how the momentum map allows one to prove the Riemannian submersion properties. The obtained geodesic equations turn out to be vector versions of the Burgers equations.

Paper Structure

This paper contains 18 sections, 19 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Optimal transport and semidirect products
  3. Geometric setting of scalar optimal transport
  4. Geodesics in the scalar transport
  5. Semi-direct products
  6. Mass transport of vector half-densities
  7. The group action on vector half-densities
  8. Riemannian metrics and submersion
  9. Geodesics and Hamiltonian equations
  10. Unbalanced mass transport
  11. Various settings of the unbalanced transport
  12. Vector unbalanced transport
  13. Universal mass transport of matrix densities
  14. Unbalanced optimal transport of matrix-valued densities
  15. Balanced matrix transport
  16. ...and 3 more sections

Key Result

Theorem 1.1

The reduced geodesic equations on the corresponding (co)tangent bundle $T^{(*)}\widetilde{G}= T^{(*)}(\operatorname{Diff}(M)\ltimes C^\infty(M,\operatorname{SO}(k)))$, corresponding to an optimal transport of vector densities, are given by where $\mathbf{v} = (\varphi,A)\cdot \mathbf{v}_0 = {v}\otimes\sqrt\varrho$. In particular, the scalar density $\varrho$ defined by $\varrho = \lvert\mathbf{v}

Figures (2)

  • Figure 1: Illustration of the relation between the group, the vector half-densities, and the densities. The projections, first by the action of $\widetilde{G}$ on $\mathbf v_0$ and then by taking the squared norm of the vector half-density, give rise to Riemannian submersions.
  • Figure 2: Commutative diagram for the relation between spaces for scalar, vector, and matrix valued optimal transport, in both balanced (upper) and unbalanced (lower) settings. (More precisely, the map from vector densities to matrix ones takes the former to positive semi-definite matrices ${\rm Sym}_+(k)$, and hence to the closure of $\rm MDens$, as discussed in Remark \ref{['rem:closure']}.)

Theorems & Definitions (52)

  • Theorem 1.1: = Theorem \ref{['prop:governing_abstract']}$'$
  • Theorem 1.2: = Theorem \ref{['thm:governing_equations_MDens_norm']}$'$
  • Theorem 2.1: Otto2001
  • Proposition 2.2: Otto2001
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 42 more