This thesis uses Kantorovich-Rubinstein distance for classifying points based on their measures.
problem Classifying points based on their measures in a metric space.
method Using Kantorovich-Rubinstein distance as a metric in the space of measures to capture geometry and topology.
result A large Kantorovich-Rubinstein distance indicates the existence of a 1-Lipschitz classifier that well classifies the points.
Optimal transport adapted for contaminated probabilities, showing equivalence under specific conditions.
problem Adapting optimal transport for ε ε ε -contaminated sets. method Generalized optimal transport problems with lower probabilities, showing equivalence under ε ε ε -contaminations. result Monge's and Kantorovich's problems coincide under ε ε ε -contaminated sets, but not always. Paper computes Kantorovich-Wasserstein distances on d d d -dimensional histograms efficiently.
problem Computing exact Kantorovich-Wasserstein distances between d d d -dimensional histograms. method Uses ( d + 1 ) (d+1) ( d + 1 ) -partite graph to solve as uncapacitated minimum cost flow problem. result Approach is competitive with state-of-the-art optimal transport algorithms.
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.
Study optimal transport on globally hyperbolic spacetimes, focusing on weak Kantorovich potentials' regularity.
problem Investigate regularity of weak Kantorovich potentials on globally hyperbolic spacetimes.
method Apply insights from Riemannian and Lorentzian cases to study π π π -solutions. result Conclude existence, uniqueness, and structure of optimal transport maps.
Unified approach solves Kyle model with dynamic information.
problem Solving a generalized Kyle model with dynamic information.
method Monge-Kantorovich duality and backward stochastic partial differential equations.
result Characterization of optimal strategies and pricing rules.
Solves optimal transport in Lorentz-Finsler spacetimes, generalizing previous work.
problem Optimal transport in Lorentz-Finsler geometry.
method Solves the Kantorovich and Monge problems for globally hyperbolic Lorentz-Finsler spacetimes.
result Generalizes previous results on optimal transport in spacetimes.
Optimizes angular velocity transfers for rigid bodies under deadline constraints.
problem Stochastic guidance of spin states of rigid bodies over a hard deadline.
method Structural analysis of Kantorovich optimal coupling formulation for nonlinear dynamics.
result Derives the ground cost for optimal transport of angular velocity.
Formula derived for curvature in measure spaces.
problem Deriving sectional curvature in measure spaces.
method Explicit formula derivation for sectional curvature in M ( M ) {\cal M}(M) M ( M ) with metrics H K HK H K and W 2 W_2 W 2 . result Curvature analysis in M ( M ) {\cal M}(M) M ( M ) reveals both negative and positive components. New inequalities link probability density norms to Sobolev norms and Kantorovich distances.
problem Bounding probability density norms on smooth weighted Riemannian manifolds.
method Refining and generalizing interpolation inequalities under C D ( 0 , ∞ ) CD(0, \infty) C D ( 0 , ∞ ) condition. result Established new inequalities linking L p L^p L p norms to Sobolev norms and Kantorovich distances. Proves uniqueness of barycenters on manifolds without restrictions.
problem Finding unique barycenters on complex geometric spaces.
method Introduces new disintegrated Monge-Kantorovich metrics for barycenter problems.
result Uniqueness of barycenters on connected, complete Riemannian manifolds.
This paper studies neural network operators and their convergence properties.
problem Understanding the approximation and convergence of neural network operators.
method Proves density results, convergence estimates, and Voronovskaya-type theorems.
result Establishes quantitative convergence estimates and derives Voronovskaya-type theorems.
New method calculates cut locus on Riemannian manifolds using optimal transport.
problem Computing the cut locus on compact Riemannian manifolds.
method Characterization via optimal transport density solution of Monge-Kantorovich equations, numerical approximation.
result Proposed novel framework for numerical approximation of cut locus.
Abstract theory extends optimal transport to Banach lattices.
problem Generalize optimal transport theory to Banach lattices.
method Abstract framework, duality theory, Banach lattice, order unit.
result Characterization of dual elements for generalized optimal transport.
Dual representation of Kantorovich functional using martingale measures.
problem Representation of Kantorovich functional on Skorokhod space.
method Choquet capacity generated by martingale measures with constraints.
result Dual representation of Kantorovich functional.
Geometric interpretation of optimal transportation improves generative models.
problem Improving the efficiency and effectiveness of generative models.
method Geometric approach to optimal transportation and variational methods to construct convex polytopes.
result Optimal transportation can be simplified by optimizing the discriminator, leading to better performance.
The paper solves optimal transport problems with domain constraints.
problem Optimal transport problems with domain constraints.
method Characterizes existence of a probability measure with convex transport constraints.
result Obtains Kantorovich duality and monotonicity principle.
Unified Kantorovich duality for multimarginal optimal transport on Polish spaces.
problem Optimal transport of multiple probability distributions.
method Unified Kantorovich duality theory for multimarginal optimal transport on general Polish product spaces.
result Unified duality theory for multimarginal optimal transport, extending classical two-marginal conjugacy.
The paper establishes general results in Lorentzian optimal transport theory.
problem Establishing strong duality and optimality conditions in Lorentzian optimal transport.
method Providing non-trivial assumptions on measures, characterizing optimality, and proving regularity results.
result Regularity results for c c c -convex functions and (weak) Kantorovich potentials do not extend to the Lorentzian setting, but under suitable assumptions, they are locally semconvex. Revisits shallow neural networks using Lipschitz norms and measures.
problem Existence and compactness of minimizers in neural network formulations.
method Mean field parametrization, signed measures, duality pairings, Kantorovich-Rubinstein norms.
result Compactness results and uniform large data limits for empirical risk minimization.
Efficiently predicts optimal transport plans using sliced potentials.
problem Predicting optimal transport plans across multiple measure pairs efficiently.
method Regression-based and objective-based amortization strategies using sliced optimal transport potentials.
result Efficient and accurate prediction of optimal transport plans for various tasks.
New algorithm solves unbalanced optimal transport on trees in quasi-linear time.
problem Efficiently solving unbalanced optimal transport problems on trees.
method Proposed an algorithm that solves a more general unbalanced optimal transport problem exactly in quasi-linear time on a tree metric.
result Solves unbalanced optimal transport on trees in quasi-linear time (less than one second for a tree with one million nodes).
New algorithm for linear bandits tackles Optimal Transport problems.
problem Optimal Transport problems not covered by traditional linear bandits.
method Embed actions into a Hilbertian subspace, penalize optimism, use least-squares estimation.
result Achieves same regret bounds as OFUL but interpolates between i l d e O ( T ) ilde{\mathcal O}(\sqrt{T}) i l d e O ( T ) and O ( T ) {\mathcal O}(T) O ( T ) . Two new couplings for probability distributions are constructed and analyzed.
problem Constructing optimal couplings for two probability distributions.
method Optimizes constrained Monge-Kantorovich transport problems with supermartingales.
result Two new couplings are identified and characterized.
A general duality proof for Wasserstein distributionally robust optimization.
problem Optimizing under uncertainty with Wasserstein distance.
method One-dimensional convex analysis and interchangeability principle.
result General duality result holds for various distributions and costs.
Study non-asymptotic behavior of Coulomb gas on compact manifolds.
problem Understand the behavior of Coulomb gas on compact manifolds.
method Use Kantorovich-Wasserstein distance, empirical measure, and heat kernel.
result Prove concentration inequality in Kantorovich-Wasserstein distance.
Unified framework for analyzing gradient flows of measures with exponential decay of entropy.
problem Analyzing exponential decay of entropy functionals in gradient flows of measures.
method Characterization of global exponential decay behaviors using Hellinger-Kantorovich geometry, shape-mass decomposition, and Polyak-Łojasiewicz-type inequalities.
result Unified theoretical framework for gradient flows with complete analysis of exponential decay behaviors.
A new associative memory uses Sinkhorn divergence for efficient pattern retrieval.
problem Efficiently retrieving patterns from large datasets of weighted point clouds.
method Derived retrieval dynamics as a SHK gradient flow, discretized for a deterministic algorithm.
result Proved basin invariance, geometric convergence, and robust recovery from perturbations.
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…
The dual representation of the martingale optimal transport problem in the Skorokhod space of multi dimensional cadlag processes is proved. The dual is a minimization problem with constraints involving stochastic integrals and is similar to the Kantorovich dual of the standard optimal transport problem. The constraints…
New framework for optimal transport with jumps over intermediate spaces.
problem Optimal transport with mass jumps over intermediate spaces.
method Hierarchical Jump multi-marginal transport (HJMOT) on Polish spaces.
result Existence and uniqueness of Monge solutions under sequential differentiability and twist condition.
This work develops sampling methods for differential privacy using SHK geometry.
problem Approximating sampling for the exponential mechanism in differential privacy.
method Develops perturbation theory for SHK gradient flows and applies to differential privacy.
result Derives time-dependent Pure-DP guarantees and Approximate-DP certificates.
New algorithm for estimating multivariate quantiles using stochastic optimal transport.
problem Estimating multivariate quantiles from data.
method Stochastic algorithm for entropic optimal transport in Banach spaces, using Fourier coefficients.
result Almost sure convergence of the stochastic algorithm in infinite-dimensional Banach spaces.
New optimal transport divergences derived from scoring functions.
problem Developing new divergences for optimal transport.
method Using scoring functions as cost functions in optimal transport.
result Comonotonic coupling is optimal for many new divergences.
Optimizes risk measures given known marginal distributions of two unknown factors.
problem Determining an upper bound for spectral risk measures with unknown joint distribution.
method Introduces Maximum Spectral Measure (MSP) as a worst-case risk measure, formulated as an optimization problem with a more general objective function.
result Characterizes the continuity properties of the optimal value function and optimal solution set with respect to marginal distributions.
HMC proves contractive for multimodal distributions.
problem Hamiltonian Monte Carlo (HMC) convergence for multimodal distributions.
method Coupling approach to prove contractive step w.r.t. Kantorovich distance.
result Explicit bounds for HMC convergence to stationary distribution.
New models improve classification model performance, especially robust to small training sets.
problem Improving classification model performance, especially robust to small training sets.
method Distributionally robust AUC maximization models using Kantorovich metric and hinge loss function.
result The proposed DR-AUC models outperform standard models in general and worst-case out-of-sample performance.
Optimal Transport CycleGAN improves unsupervised learning in imaging problems.
problem Improving unsupervised learning in inverse problems using generative models.
method Developed an OT-cycleGAN architecture using a PLS cost with deep learning-based inverse path penalty.
result Distinct variations of cycleGAN architecture can be derived based on forward problem knowledge.
Study sharp convergence rates of empirical UOT for spatio-temporal point processes.
problem Statistical analysis of UOT for spatio-temporal point processes.
method Empirical plug-in estimators for Kantorovich-Rubinstein distance between intensity measures.
result Sharp convergence rates of empirical UOT in terms of intrinsic dimensions of measures.
New framework enhances neural network robustness against adversarial attacks.
problem Vulnerability of deep neural networks to small perturbations.
method Integrates Lipschitz constraint using optimal transport and hinge regularization.
result Proposes a new loss function that certifies adversarial robustness.
New information geometry from exponentially concave functions.
problem Optimal transport and information geometry.
method Information geometry tools applied to exponentially concave functions.
result New information geometry on the unit simplex.
Some optimization or equilibrium problems involving somehow the concept of optimal transport are presented in these notes, mainly devoted to applications to economic and game theory settings. A variant model of transport, taking into account traffic congestion effects is the first topic, and it shows various links with…
This paper uses UOT metrics for better dimensionality reduction and classification/clustering.
problem Improving dimensionality reduction and classification/clustering methods.
method Uses Hellinger--Kantorovich metric from unbalanced optimal transport (UOT).
result UOT outperforms Euclidean and OT-based methods in classification and clustering tasks.
The study analyzes the evolution of Gaussian measures under a specific gradient flow.
problem Analyzing the evolution of Gaussian measures under a specific gradient flow.
method Derives ordinary differential equations governing the evolution of mean, covariance, and mass under the HK-Boltzmann gradient flow.
result Exponential convergence to equilibrium demonstrated through Polyak-Lojasiewicz-type inequalities.
Study robust distribution estimation with Wasserstein distance, achieving optimal risk.
problem Robust distribution estimation under adversarial corruption.
method Combining partial OT and minimum distance estimation, proving structural properties and deriving a novel dual form.
result Achieves minimax-optimal robust estimation risk in many settings.
The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the…
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 $…
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…