13 research outputs found
Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm
Starting from Brenier's relaxed formulation of the incompressible Euler
equation in terms of geodesics in the group of measure-preserving
diffeomorphisms, we propose a numerical method based on Sinkhorn's algorithm
for the entropic regularization of optimal transport. We also make a detailed
comparison of this entropic regularization with the so-called Bredinger
entropic interpolation problem. Numerical results in dimension one and two
illustrate the feasibility of the method
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
A dynamical--topological obstruction for smooth isometric embeddings of Riemannian manifolds via incompressible Euler equations
We obtain a dynamical--topological obstruction for the existence of isometric
embedding of a Riemannian manifold-with-boundary : if the first real
homology of is nontrivial, if the centre of the fundamental group is
trivial, and if is isometrically embedded into a Euclidean space of
dimension at least , then the isometric embedding must violate a certain
dynamical, kinetic energy-related condition (the "rigid isotopy extension
property" in Definition 1.1). The arguments are motivated by the incompressible
Euler equations with prescribed initial and terminal configurations in
hydrodynamics
Convergence rate of entropy-regularized multi-marginal optimal transport costs
We investigate the convergence rate of multi-marginal optimal transport costs
that are regularized with the Boltzmann-Shannon entropy, as the noise parameter
tends to . We establish lower and upper bounds on the
difference with the unregularized cost of the form
for some explicit dimensional
constants depending on the marginals and on the ground cost, but not on the
optimal transport plans themselves. Upper bounds are obtained for Lipschitz
costs or locally semi-concave costs for a finer estimate, and lower bounds for
costs satisfying some signature condition on the mixed second
derivatives that may include degenerate costs, thus generalizing results
previously in the two marginals case and for non-degenerate costs. We obtain in
particular matching bounds in some typical situations where the optimal plan is
deterministic
Second order models for optimal transport and cubic splines on the Wasserstein space
On the space of probability densities, we extend the Wasserstein geodesics to
the case of higher-order interpolation such as cubic spline interpolation.
After presenting the natural extension of cubic splines to the Wasserstein
space, we propose a simpler approach based on the relaxation of the variational
problem on the path space. We explore two different numerical approaches, one
based on multi-marginal optimal transport and entropic regularization and the
other based on semi-discrete optimal transport