arXiv research

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

New framework for optimal transport with jumps over intermediate spaces.

Optimizes angular velocity transfers for rigid bodies under deadline constraints.

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

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

Formula derived for curvature in measure spaces.

This paper studies neural network operators and their convergence properties.

Efficiently predicts optimal transport plans using sliced potentials.

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.

Unified approach solves Kyle model with dynamic information.

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,…

New framework enhances neural network robustness against adversarial attacks.

Revisits shallow neural networks using Lipschitz norms and measures.

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

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…

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

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…

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

Unified Kantorovich duality for multimarginal optimal transport on Polish spaces.

The paper establishes general results in Lorentzian optimal transport theory.

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

New algorithm for estimating multivariate quantiles using stochastic optimal transport.

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.

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…

The computer program SnapPea can approximate whether or not a three manifold whose boundary consists of tori has a complete hyperbolic structure, but it can not prove conclusively that this is so. This article provides a method for proving that such a manifold has a complete hyperbolic structure based on the approximat…

A general duality proof for Wasserstein distributionally robust optimization.

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 algorithm for linear bandits tackles Optimal Transport problems.

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…

We prove that, if $Ω\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial Ω$ of the function $\varphi(y) = \int_0^{λ(y)} \prod_{j=1}^{n-1}[1-t κ_j(y)]\, dt$ implies that $Ω$ is a ball. Here $k_j(y)$ and $λ(y)$ denote respectively the principal curvatures and the cut v…

New optimal transport divergences derived from scoring functions.

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…

Muon dynamics study uses spectral Wasserstein flow for optimization stability.

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

New Fourier metrics equivalent to Wasserstein distances in image processing.

Generative sampler learns velocity fields for efficient posterior inference.

In this paper we investigate model-independent bounds for exotic options written on a risky asset. Based on arguments from the theory of Monge-Kantorovich mass-transport we establish a dual version of the problem that has a natural financial interpretation in terms of semi-static hedging. In particular we prove that th…

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

Let $M,N$ be two smooth compact hypersurfaces of $\mathbb{R}^n$ which bound strictly convex domains equipped with two absolutely continuous measures $μ$ and $ν$ (with respect to the volume measures of $M$ and $N$). We consider the optimal transportation from $μ$ to $ν$ for the quadratic cost. Let $(φ:m \to \mathbb{R},ψ…

Study robust distribution estimation with Wasserstein distance, achieving optimal risk.

We study the problem of stopping a Brownian motion at a given distribution $ν$ while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set $\mathcal{T}(ν)$ of stopping times embedding $ν$ is weakly dense in the set $\mathc…

New stability theory for Sinkhorn semigroups with explicit decay rates.