AI 中文总结
本文提出容量约束的Wasserstein重心问题，证明其存在性与强对偶，并引入熵正则化，获得唯一解及容量截断吉布斯公式，且正则化解收敛至最小熵解。
AI 中文摘要
我们引入了一个容量约束的Wasserstein重心问题，其中从输入测度到重心的每个传输计划都受限于一个预设的容量测度。在紧致域上，我们证明了容量约束重心的存在性，并建立了强对偶公式。随后，在有界容量密度假设下，我们研究了熵正则化版本。正则化问题具有唯一的传输计划元组和唯一重心，而其对偶问题涉及显式的封顶指数惩罚。只要对偶最大化器存在，最优密度满足容量截断的吉布斯公式。最后，我们证明了最优值的$O(\epsilon)$估计，并表明正则化计划收敛到未正则化问题的最小熵最优解。
英文摘要
We introduce a capacity-constrained Wasserstein barycenter problem in which each transport plan from an input measure to the barycenter is bounded by a prescribed capacity measure. On compact domains, we prove existence of capacity-constrained barycenters and establish a strong duality formula. We then study an entropy-regularized version under bounded capacity-density assumptions. The regularized problem has a unique optimal tuple of transport plans and a unique barycenter, while its dual involves an explicit capped-exponential penalty. Whenever dual maximizers exist, the optimal densities satisfy a capacity-clipped Gibbs formula. Finally, we prove an $O(ε)$ estimate for the optimal values and show that the regularized plans converge to the minimum-entropy optimal solution of the unregularized problem.
