Wasserstein Gradients for the Temporal Evolution of Probability Distributions
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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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