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Wasserstein Gradients for the Temporal Evolution of Probability Distributions

by   Yaqing Chen, et al.

Many studies have been conducted on flows of probability measures, often in terms of gradient flows. We introduce here a novel approach for the modeling of the instantaneous evolution of empirically observed distribution flows over time with a data-analytic focus that has not yet been explored. The proposed model describes the observed flow of distributions on one-dimensional Euclidean space R over time based on the Wasserstein distance, utilizing derivatives of optimal transport maps over time. The resulting time dynamics of optimal transport maps are illustrated with time-varying distribution data that include yearly income distributions, the evolution of mortality over calendar years, and data on age-dependent height distributions of children from the longitudinal Zürich growth study.


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