论文标题
通用的barycenters and差异最大化公制空间
Generalized barycenters and variance maximization on metric spaces
论文作者
论文摘要
我们表明，概率度量$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ x $的完整公制空间$ m $的差异受$ x $ $ x $ $ x $ $ p（m）$ m $ m $ M $的$ x $的clyradius $ r $的平方，配备了Wasserstein Metric。当$ x $上的措施的重中心是唯一的（例如猫（$ 0 $）的空间）时，我们的方法表明，$ r $实际上与周面$ x $相吻合，因此此结果扩大了欧几里得空间的林·米肯（Lim-McCann）的最新结果。我们的方法涉及$ p（x）\ times p（m）$上的双线性最小值理论，并轻松扩展到方差被非常一般的时刻所取代时。作为一个应用程序，我们提供了Jung定理关于CAT（$ K $）空间的简单证明，这最初是由于Dekster和Lang-Schroeder所致。
We show that the variance of a probability measure $μ$ on a compact subset $X$ of a complete metric space $M$ is bounded by the square of the circumradius $R$ of the canonical embedding of $X$ into the space $P(M)$ of probability measures on $M$, equipped with the Wasserstein metric. When barycenters of measures on $X$ are unique (such as on CAT($0$) spaces), our approach shows that $R$ in fact coincides with the circumradius of $X$ and so this result extends a recent result of Lim-McCann from Euclidean space. Our approach involves bi-linear minimax theory on $P(X) \times P(M)$ and extends easily to the case when the variance is replaced by very general moments. As an application, we provide a simple proof of Jung's theorem on CAT($k$) spaces, a result originally due to Dekster and Lang-Schroeder.
