arXiv research

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

Formula derived for curvature in measure spaces.

Proves uniqueness of barycenters on manifolds without restrictions.

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

This paper studies neural network operators and their convergence properties.

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

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…

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

This study examines how well GANs estimate the Wasserstein metric.

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

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 show that for a metric space with an even number of points there is a 1-Lipschitz map to a tree-like space with the same matching number. This result gives the first basic version of an unoriented Kantorovich duality. The study of the duality gives a version of global calibrations for 1-chains with coefficients in $…

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

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.

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

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,…

Revisits shallow neural networks using Lipschitz norms and measures.

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

This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, …

New Fourier metrics equivalent to Wasserstein distances in image processing.

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

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

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…

Unified Kantorovich duality for multimarginal optimal transport on Polish spaces.

The paper establishes general results in Lorentzian optimal transport theory.

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

New framework for optimal transport with jumps over intermediate spaces.

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.

Efficiently predicts optimal transport plans using sliced potentials.

This paper develops efficient bounds on the Wasserstein metric for discrete measures.

Generative modeling on metric graphs using neural optimal transport

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 …

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…

We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our ma…

New methods improve stability of Sinkhorn algorithm in machine learning.

New algorithm for linear bandits tackles Optimal Transport problems.

Geometric analysis on diffeomorphism groups for fluid dynamics and information geometry.

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 neural method calculates EMD for particle physics data.

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…

Proposes a new distance metric for multi-marginal optimal transport.

New optimal transport divergences derived from scoring functions.

Muon dynamics study uses spectral Wasserstein flow for optimization stability.