arXiv research

Study optimal transport on globally hyperbolic spacetimes, focusing on weak Kantorovich potentials' regularity.

Efficiently predicts optimal transport plans using sliced potentials.

The paper establishes general results in Lorentzian optimal transport theory.

Unified approach solves Kyle model with dynamic information.

Unified Kantorovich duality for multimarginal optimal transport on Polish spaces.

This work develops sampling methods for differential privacy using SHK geometry.

New algorithm for estimating multivariate quantiles using stochastic optimal transport.

In this work, we show the intrinsic relations between optimal transportation and convex geometry, especially the variational approach to solve Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes. This leads to a geometric interpretation to generative models, and leads to a novel …

New method trains normalizing flows using entropy-regularized transport.

This thesis uses Kantorovich-Rubinstein distance for classifying points based on their measures.

A new associative memory uses Sinkhorn divergence for efficient pattern retrieval.

Formula derived for curvature in measure spaces.

We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier's theorem of existence of optimal ma…

This paper studies neural network operators and their convergence properties.

Generative model improved using Liouville PDE-based sliced-Wasserstein flow.

Optimal transport adapted for contaminated probabilities, showing equivalence under specific conditions.

The paper is accompanying "A general Duality Theorem for the Monge-Kantorovich Transport Problem". We explain the methods used in this article in an elementary setting and present two examples complementing the results obtained therein.

We refine and generalize several interpolation inequalities bounding the $L^p$ norm of a probability density with respect to the reference measure $μ$ by its Sobolev norm and the Kantorovich distance to $μ$ on a smooth weighted Riemannian manifold satisfying $CD(0, \infty)$ condition.

Proves uniqueness of barycenters on manifolds without restrictions.

This paper presents a novel method to compute the exact Kantorovich-Wasserstein distance between a pair of $d$-dimensional histograms having $n$ bins each. We prove that this problem is equivalent to an uncapacitated minimum cost flow problem on a $(d+1)$-partite graph with $(d+1)n$ nodes and $dn^{\frac{d+1}{d}}$ arcs,…

Optimizes angular velocity transfers for rigid bodies under deadline constraints.

Revisits shallow neural networks using Lipschitz norms and measures.

Based on a new coupling approach, we prove that the transition step of the Hamiltonian Monte Carlo algorithm is contractive w.r.t. a carefully designed Kantorovich (L1 Wasserstein) distance. The lower bound for the contraction rate is explicit. Global convexity of the potential is not required, and thus multimodal targ…

New stability theory for Sinkhorn semigroups with explicit decay rates.

Let (X,L) be a (semi-) polarized complex projective variety and T a real torus acting holomorphically on X with moment polytope P. Given a probability density g on P we introduce a new type of Monge-Ampere measure on X, defined for singular T-invariant metrics on the line bundle L, generalizing the ordinary Monge-Amper…

New method calculates cut locus on Riemannian manifolds using optimal transport.

Unified framework for analyzing gradient flows of measures with exponential decay of entropy.

Brenier isotonic regression extends multi-output isotonic regression using optimal transport.

We prove the following nonholonomic version of the classical Moser theorem: given a bracket-generating distribution on a connected compact manifold (possibly with boundary), two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution. We describe formal solutions of…

New framework for optimal transport with jumps over intermediate spaces.

We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form of the optimal transportation problem, the Brenier theorem states that the optim…

Generative modelling is often cast as minimizing a similarity measure between a data distribution and a model distribution. Recently, a popular choice for the similarity measure has been the Wasserstein metric, which can be expressed in the Kantorovich duality formulation as the optimum difference of the expected value…

Imitation Learning describes the problem of recovering an expert policy from demonstrations. While inverse reinforcement learning approaches are known to be very sample-efficient in terms of expert demonstrations, they usually require problem-dependent reward functions or a (task-)specific reward-function regularizatio…

In this paper, we present a novel and principled approach to learn the optimal transport between two distributions, from samples. Guided by the optimal transport theory, we learn the optimal Kantorovich potential which induces the optimal transport map. This involves learning two convex functions, by solving a novel mi…

Study sharp convergence rates of empirical UOT for spatio-temporal point processes.

This paper uses UOT metrics for better dimensionality reduction and classification/clustering.

The study analyzes the evolution of Gaussian measures under a specific gradient flow.

A general duality proof for Wasserstein distributionally robust optimization.

New algorithm for linear bandits tackles Optimal Transport problems.

Optimal transport improves multivariate prediction uncertainty quantification.

Method learns conditional distributions using neural entropic optimal transport.

Paper develops a generative model using Wasserstein-2 loss.

New algorithm solves unbalanced optimal transport on trees in quasi-linear time.

The optimal transport problem is studied in the context of Lorentz-Finsler geometry. For globally hyperbolic Lorentz-Finsler spacetimes the first Kantorovich problem and the Monge problem are solved. Further the intermediate regularity of the transport paths is studied. These results generalize parts of Bertrand & Puel…

Paper introduces SGA for barycenter optimization in optimal transport.

New optimal transport divergences derived from scoring functions.

Muon dynamics study uses spectral Wasserstein flow for optimization stability.

New models improve classification model performance, especially robust to small training sets.