Optimal transport and barycenters for dendritic measures

TL;DR

This paper introduces a new variant of the Wasserstein distance tailored for dendritic measures, enabling meaningful comparison and interpolation of plant root systems, with proven properties like barycenter characterization and geodesic convexity.

Contribution

The paper develops a novel Wasserstein-based metric for dendritic measures, characterizes its barycenters, and proves geodesic convexity, addressing limitations of conventional Wasserstein methods.

Findings

Barycenters with respect to the new metric are also dendritic.

Interpolations between root-like measures preserve dendritic structure.

The metric exhibits geodesic convexity for biologically relevant functionals.

Abstract

We introduce and study a variant of the Wasserstein distance on the space of probability measures, specially designed to deal with measures whose support has a dendritic, or treelike structure with a particular direction of orientation. Our motivation is the comparison of and interpolation between plants' root systems. We characterize barycenters with respect to this metric, and establish that the interpolations of root-like measures, using this new metric, are also root like, in a certain sense; this property fails for conventional Wasserstein barycenters. We also establish geodesic convexity with respect to this metric for a variety of functionals, some of which we expect to have biological importance.