Covariance-modulated optimal transport and gradient flows

Abstract

We study a variant of the dynamical optimal transport problem 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. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed.

90 pages, 4 figures. Final revision. Published version open access linked below

References in corpus (14)

• Optimal whitening and decorrelation
• A new class of transport distances
• Ensemble Kalman Inversion: A Derivative-Free Technique For Machine Learning Tasks
• Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion
• Eulerian calculus for the displacement convexity in the Wasserstein distance
• Ensemble Kalman Sampler: mean-field limit and convergence analysis
• Transport distances and geodesic convexity for systems of degenerate diffusion equations
• On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobility
• On a constrained 2-D Navier-Stokes equation
• Is affine invariance well defined on SPD matrices? A principled continuum of metrics
• Gradient flow structure and convergence analysis of the ensemble Kalman inversion for nonlinear forward models
• Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility
• Gradient flow formulation and longtime behaviour of a constrained Fokker-Planck equation
• On a novel gradient flow structure for the aggregation equation