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On the convergence of discrete dynamic unbalanced transport models

Bowen Li, Jun Zou

TL;DR

The paper tackles the convergence of discretized matrix-valued unbalanced OT models WB$_\Lambda$ by developing an abstract Lax-equivalence–style framework and a FEM‑inspired concrete discretization. It proves a general convergence theorem ensuring weak convergence of discrete minimizers to a continuous WB$_\Lambda$ minimizer under verifiable consistency, stability, and approximability conditions, and provides a concrete scheme with proven convergence. In the Wasserstein–Fisher–Rao special case, AC requirements on endpoints can be removed via a static formulation, with quantitative endpoint error bounds and discrete-energy estimates. Collectively, these results furnish a rigorous numerical foundation for solving matrix-valued unbalanced OT problems in applications such as imaging, quantum dynamics, and tensor-field processing, enabling reliable discretization-based computation of WB$_\Lambda$ distances and their geodesics.

Abstract

A generalized unbalanced optimal transport distance ${\rm WB}_Λ$ on matrix-valued measures $\mathcal{M}(Ω,\mathbb{S}_+^n)$ was defined in [arXiv:2011.05845] à la Benamou-Brenier, which extends the Kantorovich-Bures and the Wasserstein-Fisher-Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with ${\rm WB}_Λ$. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, under the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Moreover, thanks to the static formulation, we show that such an assumption can be removed for the Wasserstein-Fisher-Rao distance.

On the convergence of discrete dynamic unbalanced transport models

TL;DR

The paper tackles the convergence of discretized matrix-valued unbalanced OT models WB by developing an abstract Lax-equivalence–style framework and a FEM‑inspired concrete discretization. It proves a general convergence theorem ensuring weak convergence of discrete minimizers to a continuous WB minimizer under verifiable consistency, stability, and approximability conditions, and provides a concrete scheme with proven convergence. In the Wasserstein–Fisher–Rao special case, AC requirements on endpoints can be removed via a static formulation, with quantitative endpoint error bounds and discrete-energy estimates. Collectively, these results furnish a rigorous numerical foundation for solving matrix-valued unbalanced OT problems in applications such as imaging, quantum dynamics, and tensor-field processing, enabling reliable discretization-based computation of WB distances and their geodesics.

Abstract

A generalized unbalanced optimal transport distance on matrix-valued measures was defined in [arXiv:2011.05845] à la Benamou-Brenier, which extends the Kantorovich-Bures and the Wasserstein-Fisher-Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with . We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, under the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Moreover, thanks to the static formulation, we show that such an assumption can be removed for the Wasserstein-Fisher-Rao distance.
Paper Structure (10 sections, 11 theorems, 190 equations, 1 table)

This paper contains 10 sections, 11 theorems, 190 equations, 1 table.

Table of Contents

  1. Introduction
  2. Transport distances on matrix-valued measures
  3. Abstract convergence framework
  4. Approximation of Radon measures
  5. Discrete matrix-valued OT problem
  6. Convergence analysis
  7. Concrete numerical scheme
  8. Convergence of discrete WFR metric
  9. Concluding remarks
  10. Proof of Proposition \ref{['prop:regular_ep']}

Key Result

Proposition 2.2

Let $\mu = \mathsf{(G,q,R)} \in \mathcal{CE}_\infty([0,1];\mathsf{G}_0,\mathsf{G}_1)$ with $\mathsf{G}_0, \mathsf{G}_1 \in \mathcal{M}(\Omega,\mathbb{S}_+^n)$. Suppose the assumptions 1 and 2 hold. Then, for small $\varepsilon > 0$, there exist $\mu^\varepsilon = (\mathsf{G}^\varepsilon,\mathsf{q}^\

Theorems & Definitions (32)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 3.1: Approximation of measure
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Definition 3.5: Spatial discrete modules
  • Remark 3.6
  • ...and 22 more