arXiv research

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

Dual representation of Kantorovich functional using martingale measures.

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

Revisits shallow neural networks using Lipschitz norms and measures.

Formula derived for curvature in measure spaces.

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

This paper studies neural network operators and their convergence properties.

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.

New optimal transport divergences derived from scoring functions.

Unified approach solves Kyle model with dynamic information.

The paper establishes general results in Lorentzian optimal transport theory.

Proves uniqueness of barycenters on manifolds without restrictions.

Efficiently predicts optimal transport plans using sliced potentials.

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

Unified Kantorovich duality for multimarginal optimal transport on Polish spaces.

Optimizes angular velocity transfers for rigid bodies under deadline constraints.

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,…

New algorithm for linear bandits tackles Optimal Transport problems.

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…

A general duality proof for Wasserstein distributionally robust optimization.

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

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…

Optimizes risk measures given known marginal distributions of two unknown factors.

New method trains normalizing flows using entropy-regularized transport.

This study examines how well GANs estimate the Wasserstein metric.

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

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

New framework for optimal transport with jumps over intermediate spaces.

New framework enhances neural network robustness against adversarial attacks.

New algorithm for estimating multivariate quantiles using stochastic optimal transport.

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

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.

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…

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 …

In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recen…

Develops theory for conditional optimal transport in infinite-dimensional spaces.

Describing shapes by suitable measures in object segmentation, as proposed in [24], allows to combine the advantages of the representations as parametrized contours and indicator functions. The pseudo-Riemannian structure of optimal transport can be used to model shapes in ways similar as with contours, while the Kanto…

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

We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequalit…

In this work, we present a method to compute the Kantorovich-Wasserstein distance of order one between a pair of two-dimensional histograms. Recent works in Computer Vision and Machine Learning have shown the benefits of measuring Wasserstein distances of order one between histograms with $n$ bins, by solving a classic…

Two probability distributions $μ$ and $ν$ in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are al…

Muon dynamics study uses spectral Wasserstein flow for optimization stability.

A new method uses optimal transport for imitation learning, improving efficiency and performance.

Generative sampler learns velocity fields for efficient posterior inference.