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This lecture presents "Covariance-Modulated Optimal Transport Geometry" by Franca Hoffmann from the California Institute of Technology, recorded at IPAM's Dynamics of Density Operators Workshop at UCLA on April 29, 2025. Explore a variant of the dynamical optimal transport problem where the energy to be minimized is modulated by the covariance matrix of the current distribution. Discover how these transport metrics naturally emerge in mean-field limits of certain ensemble Kalman methods for solving inverse problems. Learn about the splitting of the transport problem into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve, and another for its shape. Understand how gradient flows exhibit a similar splitting between moments and shapes of the distribution, demonstrating improved convergence properties compared to classical Wasserstein metrics, with exponential convergence rates independent of the Gaussian target. The presentation runs for 42 minutes.
