The problem of comparing probability distributions is at the heart of many tasks in statistics and machine learning and the most classical comparison methods assume that the distributions occur in spaces of the same dimension. Recently, a new geometric solution has been proposed to address this problem when the measures live in Euclidean spaces of differing dimensions. Here, we study the same problem of comparing probability distributions of different dimensions in the tropical geometric setting, which is becoming increasingly relevant in computations and applications involving complex, geometric data structures. Specifically, we construct a Wasserstein distance between measures on different tropical projective tori - the focal metric spaces in both theory and applications of tropical geometry - via tropical mappings between probability measures. We prove equivalence of the directionality of the maps, whether starting from the lower dimensional space and mapping to the higher dimensional space or vice versa. As an important practical implication, our work provides a framework for comparing probability distributions on the spaces of phylogenetic trees with different leaf sets.

翻译：概率分布的比较是统计学与机器学习中众多任务的核心问题，最经典的比较方法假设分布存在于相同维数的空间中。近年来，针对测度存在于不同维数欧氏空间的情况，研究者提出了一种新的几何解决方案。本文在热带几何框架下研究这一不同维数概率分布比较问题——该框架在处理涉及复杂几何数据结构的计算与应用中正日益重要。具体而言，我们通过概率测度间的热带映射，在不同维数的热带射影环面（热带几何理论与应用中的核心度量空间）上构建了Wasserstein距离。我们证明了映射方向性（从低维空间映射至高维空间，或反之）的等价性。本工作的重要实际意义在于，为比较具有不同叶集合的系统发生树空间上的概率分布提供了理论框架。