André Schlichting

Preprint: Covariance-Modulated Optimal Transport and Gradient Flows

15 February 2023

Together with Martin Burger, Franca Hoffmann, Daniel Matthes and Matthias Erbar, we investigate a new dynamical optimal transport distance in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation.

Comparison of the geodesics for covariance-constrained optimal transport (red) and the normalized geodesics for the classical Wasserstein distance (blue dashed). The first and second marginal are depicted with blue circles and red stars, respectively, with size representing the mass.
Comparison of the geodesics for covariance-constrained optimal transport (red) and the normalized geodesics for the classical Wasserstein distance (blue dashed). The first and second marginal are depicted with blue circles and red stars, respectively, with size representing the mass.

Now published in Arch. Ration. Mech. Anal. 249(1), 7 (2024).

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