Wasserstein Gradients for the Temporal Evolution of Probability Distributions

Abstract

Many studies have been conducted on flows of probability measures, often in terms of gradient flows. We utilize a generalized notion of derivatives with respect to time to model the instantaneous evolution of empirically observed one-dimensional distributions that vary over time and develop consistent estimates for these derivatives. Employing local Fréchet regression and working in local tangent spaces with regard to the Wasserstein metric, we derive the rate of convergence of the proposed estimators. The resulting time dynamics are illustrated with time-varying distribution data that include yearly income distributions and the evolution of mortality over calendar years.

References in corpus (5)

• Functional data analysis for density functions by transformation to a Hilbert space
• Amplitude and phase variation of point processes
• Intrinsic Riemannian Functional Data Analysis
• Optimal transport maps in Monge-Kantorovich problem
• A survey of some recent applications of optimal transport methods to econometrics