2 research outputs found
Fast Unbalanced Optimal Transport on a Tree
This study examines the time complexities of the unbalanced optimal transport
problems from an algorithmic perspective for the first time. We reveal which
problems in unbalanced optimal transport can/cannot be solved efficiently.
Specifically, we prove that the Kantorovich Rubinstein distance and optimal
partial transport in the Euclidean metric cannot be computed in strongly
subquadratic time under the strong exponential time hypothesis. Then, we
propose an algorithm that solves a more general unbalanced optimal transport
problem exactly in quasi-linear time on a tree metric. The proposed algorithm
processes a tree with one million nodes in less than one second. Our analysis
forms a foundation for the theoretical study of unbalanced optimal transport
algorithms and opens the door to the applications of unbalanced optimal
transport to million-scale datasets.Comment: Accepted to NeurIPS 202
Tropical Optimal Transport and Wasserstein Distances
We study the problem of optimal transport in tropical geometry and define the
Wasserstein- distances in the continuous metric measure space setting of the
tropical projective torus. We specify the tropical metric -- a combinatorial
metric that has been used to study of the tropical geometric space of
phylogenetic trees -- as the ground metric and study the cases of in
detail. The case of gives an efficient computation of geodesics on the
tropical projective torus, while the case of gives a form for Fr\'{e}chet
means and a general inner product structure. Our results also provide
theoretical foundations for geometric insight a statistical framework in a
tropical geometric setting. We construct explicit algorithms for the
computation of the tropical Wasserstein-1 and 2 distances and prove their
convergence. Our results provide the first study of the Wasserstein distances
and optimal transport in tropical geometry. Several numerical examples are
provided.Comment: 33 pages, 9 figures, 3 table