A new discrete formula connects vertex and edge distributions on graphs.
problem Optimal transport on graphs with mixed vertex and edge distributions.
method Discrete transport equation and Benamou-Brenier formulation.
result Classification of all Wasserstein-1 geodesics on graphs.
Extends optimal transport to dynamic and martingale settings.
problem Dynamic and martingale relaxation of optimal transport problems.
method Extends Benamou-Brenier formula to weak optimal transport and introduces barycentric optimal transport.
result Relates barycentric optimal transport to martingale Benamou-Brenier formula.
Modified Wasserstein metric for Gaussian distributions, invariant to isometries.
problem Distance measurement for latent Gaussian distributions invariant to isometries.
method Modified Benamou-Brenier approach leading to a Procrustes Wasserstein metric.
result For Gaussian distributions, the metric reduces to Euclidean distance between eigenvalues.
In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. We suggest a Benamou-Brenier type formulation of the martingale transport proble…
New method finds closest martingale to Brownian motion.
problem Finding optimal martingale interpolating marginals.
method Martingale Sinkhorn algorithm, iterative scheme.
result Algorithm yields Bass potential in arbitrary dimension.
Geometric Bass martingales linked to Brownian motion and geometric Brownian motion.
problem Modeling continuous martingales with prescribed initial and terminal distributions.
method Developed geometric Bass martingales and established their properties.
result Explicit bijection and representation of geometric Bass martingales.
Improved tracking of tangled point sources using Riemannian metrics.
problem Tangled point source trajectories in temporal stacks.
method Lifting to higher-dimensional space of roto-translation group, new regularisation based on relaxed Reeds-Shepp metric.
result Reconstruction and untangling of trajectories even from numerical standpoint.
A new method steers Gaussian distributions with minimal effort.
problem Steering high-dimensional Gaussian distributions efficiently.
method Sliced feedback controller using one-dimensional projections and averaging.
result The method steers Gaussian distributions to targets efficiently.
Let K be an irreducible and reversible Markov kernel on a finite set X. We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wassers…
New optimal transport method handles mass creation and destruction.
problem Optimal re-balancing of portfolios with mass creation or destruction.
method Formalizes an optimal transport problem with mass-change factor.
result Existence of optimal transport plans and maps established.
A mesh-free method solves continuum-marginal optimal transport problems.
problem Recovering minimum-energy velocity fields from time-continuous probability marginals.
method Embeds weak continuity equation in a reproducing kernel Hilbert space, optimizing with mini-batch stochastic methods.
result Accurately recovers drift and maintains marginal consistency in synthetic experiments.
Muon dynamics study uses spectral Wasserstein flow for optimization stability.
problem Optimizing deep learning models with gradient normalization.
method Introduces Spectral Wasserstein distances for matrix flows, proving equivalence with Benamou--Brenier formulation.
result Gradient-flow interpretation of mean-field normalized training dynamics.
Introduces a new divergence measure for optimal transport.
problem Optimal transport distances and information divergences.
method Infimal convolution formulation of proximal optimal transport divergence.
result Establishes connections to dynamic formulations and partial differential equations.
This work learns models for population dynamics using variational methods and higher-order quadrature.
problem Modeling population dynamics of physical systems with stochastic and mean-field effects.
method Variational problem to infer gradient fields, combining Monte Carlo sampling with higher-order quadrature rules.
result Accurate prediction of population dynamics over a wide range of parameters.
Study approximates probability measures using structured classes of functions.
problem Approximating probability measures in Wasserstein-p distance. method Structured classes of approximators for functions in Lp(Ω), transferring to measures in Wp(Ω). result Linear rate approximation for measures with densities bounded away from zero.
Study minimax robustness in statistical estimation under Wasserstein contamination.
problem Adversarial perturbations in statistical data.
method Developed minimax theory for ℓqr losses under Wasserstein-r contaminations. result Exact minimax risk identified for joint contaminations in location estimation and prediction in linear regression.
The paper constructs a Dirichlet form and proves functional inequalities for a specific measure.
problem Investigating functional inequalities for a specific measure in a configuration space.
method Constructing a strongly local symmetric Dirichlet form on the configuration space and proving various inequalities.
result The Dirichlet form satisfies the Bakry-Émery gradient estimate with K=0 and yields various functional inequalities. This work explores gradient flows and Riemannian structure in Gromov-Wasserstein geometry for data with global structure.
problem Suitable geometry for tasks requiring preservation of global data structure.
method Study of gradient flows and Riemannian structure in Gromov-Wasserstein geometry for distributions on \(\mathbb{R}^d\).
result Established a Benamou-Brenier-like formula for IGW and derived the IGW gradient.
Large Sinkhorn couplings improve flow models in data generation tasks.
problem Training flow models with optimal transport couplings.
method Using large batches of source and target points, and applying entropic regularization with a low ε. result Flow models perform better with large Sinkhorn couplings and low regularization.