Probability Metrics for Tropical Spaces of Different Dimensions
Roan Talbut, Daniele Tramontano, Yueqi Cao, Mathias Drton, Anthea, Monod

TL;DR
This paper introduces a novel Wasserstein distance for comparing probability distributions across different tropical spaces, enabling applications like phylogenetic tree analysis with varying leaf sets.
Contribution
It develops a tropical geometry-based framework for measuring differences between measures on spaces of different dimensions, extending existing methods.
Findings
Proves the equivalence of directionality in tropical mappings.
Provides a practical computational approach for real data.
Enables comparison of phylogenetic trees with different leaf sets.
Abstract
The problem of comparing probability distributions is at the heart of many tasks in statistics and machine learning. Established comparison methods treat the standard setting that the distributions are supported in the same space. Recently, a new geometric solution has been proposed to address the more challenging problem of comparing measures in Euclidean spaces of differing dimensions. Here, we study the same problem of comparing probability distributions of different dimensions in the tropical setting, which is becoming increasingly relevant in applications involving complex data structures such as phylogenetic trees. 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…
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Taxonomy
TopicsLeaf Properties and Growth Measurement · Data Management and Algorithms · Polynomial and algebraic computation
