熵最优传输不一定选择零温度极限
Entropic optimal transport need not select a zero-temperature limit
AI总结：
构建具有特定性质的紧致度量空间，证明熵最优传输极小值无零温度弱极限，给出紧致性定理及收敛准则，揭示成本相关性质不必然导致零温度收敛。
AI中文摘要：
我们构建了一个具有无原子概率测度和有界Lipschitz成本的紧致度量空间，其熵最优传输极小值没有零温度弱极限。具体而言，当\(\varepsilon\downarrow0\)时，\(P_\varepsilon\)不收敛。在该示例中，每个未正则化的极小值相对于\(\mu\otimes\mu\)是奇异的，使得最优面上的熵恒为\(+\infty\)。我们通过\(\operatorname{Clust}(P_\varepsilon)=\{P_w:w\in\mathcal W\}\)描述聚类集，其中\(P_w\)是两个零成本图耦合的混合，权重为\(w\)，\(\mathcal W\subset[0,1]\)是一个非退化紧致区间。然后我们计算出该区间内的两个显式点\(w^-<w^+\)。这表明成本的紧致性、无原子性和Lipschitz正则性并不意味着零温度收敛。我们还给出了一般问题的紧致性定理。如果\(C\in L^1(\mu\otimes\nu)\)在波兰空间上连续且有下界，那么零温度聚类集是最优面上的一个非空弱紧致连通子集。在证明中，我们应用了Bernton、Ghosal和Nutz的聚点定理以及\(\varepsilon\mapsto\pi_\varepsilon\)的连续性。最后，我们给出了完全收敛和聚类成员资格的局部和外部一阶准则。我们表明不收敛是可能的，但只能通过最优计划的连通连续统。
英文摘要：
We construct a compact metric space with an atomless probability measure and a bounded Lipschitz cost for which the entropic optimal-transport minimisers have no zero-temperature weak limit. More precisely, $P_\varepsilon$ does not converge as $\varepsilon\downarrow0$. In the example, every unregularised minimiser is singular with respect to $μ\otimesμ$, so that the entropy on the optimal face is identically $+\infty$. We describe the cluster set by \[ \operatorname{Clust}(P_\varepsilon)=\{P_w:w\in\mathcal W\}, \] where $P_w$ is the mixture of the two zero-cost graph couplings with weight $w$, and where $\mathcal W\subset[0,1]$ is a non-degenerate compact interval. We then compute two explicit points $w^-<w^+$ in this interval. This shows that compactness, atomlessness, and Lipschitz regularity of the cost do not imply zero-temperature convergence. We also present a compactness theorem for the general problem. If $C\in L^1(μ\otimesν)$ is continuous and bounded from below on Polish spaces, then the zero-temperature cluster set is a nonempty weakly compact connected subset of the optimal face. In the proof, we apply the cluster-point theorem of Bernton, Ghosal, and Nutz and the continuity of $\varepsilon\mapstoπ_\varepsilon$. Finally, we give local and exterior first-order criteria for full convergence and cluster membership. We show that nonconvergence is possible, but only through a connected continuum of optimal plans.
