Vol. 2, No. 3, 2020
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Optimal transport and barycenters for dendritic measures

Young-Heon Kim, Brendan Pass and David J. Schneider
Vol. 2 (2020), No. 3, 581–601
DOI: 10.2140/paa.2020.2.581
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 tree-like 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.

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Keywords
optimal transport, dendritic measures, layerwise-Wasserstein distance, plant root architecture
Mathematical Subject Classification 2010
Primary: 49K99
Secondary: 92C80
Milestones
Received: 10 October 2019
Revised: 21 February 2020
Accepted: 28 March 2020
Published: 17 November 2020
Authors
Young-Heon Kim
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada
Brendan Pass
Department of Mathematical and Statistical Sciences
University of Alberta
Edmonton, AB
Canada
David J. Schneider
Global Institute for Food Security
University of Saskatchewan
Saskatoon, SK
Canada