自由完备Wasserstein代数
Free complete Wasserstein algebras
AI总结：
本文为完备度量空间上的Wasserstein距离$W_p$提供代数公理化，并证明自由完备代数为具有有限$p$阶矩的Radon概率测度空间。
摘要：
我们提出了完备度量空间上Wasserstein距离$W_p$（$p \geq 1$）的一个代数刻画。这是编程语言中效应定量代数理论计划的一部分。特别地，我们给出了参数化于$p$的公理，用于配备概率选择运算的度量空间上的代数。这些公理表明运算构成一个重心代数，并且度量满足Wasserstein距离$W_p$的典型性质。我们证明了在完备度量空间上，此类自由完备代数是具有有限$p$阶矩的Radon概率测度空间，其度量为Wasserstein距离，运算为通常的二元凸组合。
英文摘要：
We present an algebraic account of the Wasserstein distances $W_p$ on complete metric spaces, for $p \geq 1$. This is part of a program of a quantitative algebraic theory of effects in programming languages. In particular, we give axioms, parametric in $p$, for algebras over metric spaces equipped with probabilistic choice operations. The axioms say that the operations form a barycentric algebra and that the metric satisfies a property typical of the Wasserstein distance $W_p$. We show that the free complete such algebra over a complete metric space is that of the Radon probability measures with finite moments of order $p$, equipped with the Wasserstein distance as metric and with the usual binary convex sums as operations.
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