On a general matrix valued unbalanced optimal transport and its fully discretization: dynamic formulation and convergence framework

In this work, we present a rather general class of transport distances over the space of positive semidefinite matrix valued Radon measures, called the weighted Wasserstein Bures distance, and consider the convergence property of their fully discretized counterparts. These distances are defined via a generalization of Benamou Brenier formulation of the quadratic optimal transport, based on a new weighted action functional and an abstract matricial continuity equation. It gives rise to a convex optimization problem. We shall give a complete characterization of its minimizer (i.e., the geodesic) and discuss some topological and geometrical properties of these distances. Some recently proposed models: the interpolation distance by Chen et al. [18] and the Kantorovich Bures distance by Brenier et al. [11], as well as the well studied Wasserstein Fisher Rao distance [43, 19, 40], fit in our model. The second part of this work is devoted to the numerical analysis of the fully discretization of the new transport model. We reinterpret the convergence framework proposed very recently by Lavenant [41] for the quadratic optimal transport from the perspective of Lax equivalence theorem and extend it to our general problem. In view of this abstract framework, we suggest a concrete fully discretized scheme inspired by the finite element theory, and show the unconditional convergence under mild assumptions. In particular, these assumptions are removed in the case of Wasserstein Fisher Rao distance due to the existence of a static formulation.Comment: 47 pages, raw draf

Statistical Learning in Wasserstein Space

We seek a generalization of regression and principle component analysis (PCA) in a metric space where data points are distributions metrized by the Wasserstein metric. We recast these analyses as multimarginal optimal transport problems. The particular formulation allows efficient computation, ensures existence of optimal solutions, and admits a probabilistic interpretation over the space of paths (line segments). Application of the theory to the interpolation of empirical distributions, images, power spectra, as well as assessing uncertainty in experimental designs, is envisioned

From Monge to Higgs: a survey of distance computations in noncommutative geometry

This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, and the comparison of Connes distance with the minimal length that emerges in various models of quantum spacetime. Links with other areas of mathematics are studied, in particular the horizontal distance in sub-Riemannian geometry. The interpretation of Connes distance as a noncommutative version of the Monge-Kantorovich metric in optimal transport is also discussed.Comment: Proceedings of the workshop "Noncommutative Geometry and Optimal Transport", Besan\c{c}on november 201

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations

We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal Transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized Hydrostatic-Boussinesq equations) to various models involving Optimal Transport (and the related Monge-Ampere equation. This includes the 2D semi-geostrophic equations and some fully non-linear versions of the so-called high-field limit of the Vlasov-Poisson system and of the Keller-Segel for Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology