Existence, Equivalence, and Convergence of Unbalanced Optimal Transport
Abstract
We study an unbalanced optimal transport problem with a spatially independent signed source. The target density is continuous on a bounded domain or continuous and integrable on ℝn. We establish sequential weak lower semicontinuity at the critical Sobolev regularity W1,n and existence of minimizers under prescribed distortion and Jacobian mass bounds, provided a finite-energy admissible pair exists. For domain-preserving, uniformly bi-Lipschitz affine flows, we prove an energy-preserving static-dynamic correspondence and stability of the reconstruction. Within this class, sequential Γ-convergence identifies the strong source-penalty limit as transport minimization among pairs of minimum source cost. This yields optimal-cost asymptotics and subsequential convergence of minimizers and their reconstructed paths. When a constant source is admissible, optimal sources converge strongly in L2 to the constant determined by the endpoint mass difference. We establish local constant-source feasibility for Hölder endpoint densities near a fixed reference pair and give a quantitative obstruction showing that constant-source feasibility may fail even when admissible pairs exist.
Related
- Gaussian-Restricted Barycenters for KL-Unbalanced Optimal Transport: Variational Theory and Fixed-Point Convergence
- A Network Simplex Framework for Discrete Unbalanced Optimal Transport
- Multiscale Supervised Unbalanced Optimal Transport Flow Matching
- Dynamical Unbalanced Optimal Transport: Duality, Reachability, and Computation
- One-Step Generative Modeling via Unbalanced Optimal Transport
BibTeX
@misc{zhao2026existence,
title = {{Existence, Equivalence, and Convergence of Unbalanced Optimal Transport}},
author = {Qing Zhao and Feida Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41321},
eprint = {2610.41321},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41321},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Existence, Equivalence, and Convergence of Unbalanced Optimal Transport
AU - Qing Zhao
AU - Feida Jiang
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41321
UR - https://arxiv.org/abs/2610.41321
AB - We study an unbalanced optimal transport problem with a spatially independent signed source. The target density is continuous on a bounded domain or continuous and integrable on ℝ^n. We establish sequential weak lower semicontinuity at the critical Sobolev regularity W^1,n and existence of minimizers under prescribed distortion and Jacobian mass bounds, provided a finite-energy admissible pair exists. For domain-preserving, uniformly bi-Lipschitz affine flows, we prove an energy-preserving static-dynamic correspondence and stability of the reconstruction. Within this class, sequential Γ-convergence identifies the strong source-penalty limit as transport minimization among pairs of minimum source cost. This yields optimal-cost asymptotics and subsequential convergence of minimizers and their reconstructed paths. When a constant source is admissible, optimal sources converge strongly in L^2 to the constant determined by the endpoint mass difference. We establish local constant-source feasibility for Hölder endpoint densities near a fixed reference pair and give a quantitative obstruction showing that constant-source feasibility may fail even when admissible pairs exist.
ER -