固定参考熵正则化Wasserstein重心问题的连续对偶极大元
Continuous Dual Maximizers for Fixed-Reference Entropy-Regularized Wasserstein Barycenters
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中文总结 AI 辅助
针对固定参考熵正则化Wasserstein重心问题，证明了在紧致度量空间上对偶问题存在连续极大化势函数，并建立了最优计划与重心的存在唯一性及显式密度，完善了原始-对偶理论。
中文摘要 AI 辅助
我们研究固定参考熵正则化Wasserstein重心问题，其中在优化前于重心空间上选择一个概率测度$\eta$，且第$i$个传输计划相对于$\nu_i\otimes\eta$进行正则化。早期工作为该模型建立了对偶公式，并证明了原始问题与对偶问题最优值之间的相等性。然而，值的相等性本身并不能保证对偶上确界能够达到。我们的主要结果表明，在具有连续传输代价的紧致度量空间上，对偶问题在原始连续函数类中允许存在极大化势函数。为完整性起见，我们还建立了最优传输计划元组及重心存在性与唯一性。极大化势函数通过指数原始-对偶关系恢复最优计划，并给出重心相对于$\eta$的显式密度。该密度是连续且严格正的。我们进一步证明，势函数继承代价的正则性，并且仅在自然的加法规范变换下唯一。这些结果强化了已知的值对偶理论，并为固定参考正则化重心问题提供了完整的连续原始-对偶描述。
英文摘要
We study the fixed-reference entropy-regularized Wasserstein barycenter problem, in which a probability measure $η$ on the barycenter space is chosen before optimization and the $i$th transport plan is regularized relative to $ν_i\otimesη$. Earlier work established a dual formulation for this model and equality between the primal and dual optimal values. However, equality of values does not by itself guarantee that the dual supremum is achieved. Our main result shows that, on compact metric spaces with continuous transport costs, the dual problem admits maximizing potentials in the original class of continuous functions. For completeness, we also establish existence and uniqueness of the optimal tuple of transport plans and of the barycenter. The maximizing potentials recover the optimal plans through exponential primal--dual relations and give an explicit density of the barycenter with respect to $η$. This density is continuous and strictly positive. We further show that the potentials inherit regularity from the costs and are unique up to the natural additive gauge transformations. These results strengthen the known value-duality theory and provide a complete continuous primal--dual description of the fixed-reference regularized barycenter problem.
