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.
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 CD(0,∞) condition. result Established new inequalities linking Lp norms to Sobolev norms and Kantorovich distances. Paper computes Kantorovich-Wasserstein distances on d-dimensional histograms efficiently.
problem Computing exact Kantorovich-Wasserstein distances between d-dimensional histograms. method Uses (d+1)-partite graph to solve as uncapacitated minimum cost flow problem. result Approach is competitive with state-of-the-art optimal transport algorithms.
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.
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-solutions. result Conclude existence, uniqueness, and structure of optimal transport maps.
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.
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.
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.
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.
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).
Method approximates Wasserstein distance between 2D histograms using min cost flow.
problem Computing Wasserstein distance between 2D histograms efficiently.
method Transforms the problem into an uncapacitated min cost flow problem.
result Approximates optimal solution with reduced network size O(n). Paper develops Gaussian process for distributions using Wasserstein distances.
problem Forecasting Gaussian processes indexed by probability distributions.
method Developed positive definite kernels based on Wasserstein distances.
result Efficient forecasting of Gaussian processes indexed by distributions.
New stability theory for Sinkhorn semigroups with explicit decay rates.
problem Stability and convergence of Sinkhorn iterations for various divergences.
method Operator-theoretic framework based on Lyapunov techniques.
result Explicit exponential decay rates for Sinkhorn iterates.
Survey of distance computations in noncommutative geometry, linking to optimal transport.
problem Computing distances in noncommutative geometry.
method Review of explicit computations in various noncommutative geometries.
result Connes distance as a noncommutative version of Monge-Kantorovich metric.
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.
Formula derived for curvature in measure spaces.
problem Deriving sectional curvature in measure spaces.
method Explicit formula derivation for sectional curvature in M(M) with metrics HK and W2. result Curvature analysis in M(M) reveals both negative and positive components. 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.
Adapts Stein's method for geometric inequalities, addressing boundary terms.
problem Geometric inequalities and their stability under constraints.
method Uses elliptic PDE with oblique boundary condition to handle boundary terms.
result Stability results for various geometric inequalities with respect to a new distance.
New Fourier metrics equivalent to Wasserstein distances in image processing.
problem Equivalence of Fourier-based and Wasserstein metrics in imaging problems.
method Extensions of Fourier-based metrics to handle different centers of mass and discrete measures, showing equivalence to Wasserstein distances.
result New Fourier metrics are equivalent to Wasserstein distances with explicit constants, improving runtime in image processing.
Optimal transport adapted for contaminated probabilities, showing equivalence under specific conditions.
problem Adapting optimal transport for ε-contaminated sets. method Generalized optimal transport problems with lower probabilities, showing equivalence under ε-contaminations. result Monge's and Kantorovich's problems coincide under ε-contaminated sets, but not always. Muon dynamics study uses spectral Wasserstein flow for optimization stability.
problem Optimizing deep learning models with gradient normalization.
method Introduces Spectral Wasserstein distances for matrix flows, proving equivalence with Benamou--Brenier formulation.
result Gradient-flow interpretation of mean-field normalized training dynamics.
This work robustifies Wasserstein distance estimation with MoM estimators for outlier-polluted data.
problem Estimating Wasserstein distance between two distributions with outliers.
method Introducing MoM-based robust estimators for Wasserstein distance.
result Consistent MoM-based estimators for Wasserstein distance with convergence rates.
Proposes a new distance metric for multi-marginal optimal transport.
problem Computational scalability in multi-marginal optimal transport.
method Random one-dimensional projections to construct sliced multi-marginal Wasserstein distance.
result Sliced multi-marginal Wasserstein distance is a metric with dimension-free sample complexity.
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.
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.
New methods improve stability of Sinkhorn algorithm in machine learning.
problem Stability of Sinkhorn semigroups in high-dimensional settings.
method Semigroup analysis based on contraction coefficients and Lyapunov-type operator-theoretic techniques.
result Unified and simplified arguments in Sinkhorn algorithm stability.
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.
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.
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.
We prove that, if Ω⊂Rn is an open bounded starshaped domain of class C2, the constancy over ∂Ω of the function φ(y)=∫0λ(y)∏j=1n−1[1−tκj(y)]dt implies that Ω is a ball. Here kj(y) and λ(y) denote respectively the principal curvatures and the cut v…
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.
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.
New neural method calculates EMD for particle physics data.
problem Metric for particle collider events based on Wasserstein metric.
method Neural network architecture estimating EMD using Kantorovich-Rubinstein duality.
result Differentiable way to calculate EMD for geometric fitting.
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.
Paper relaxes the Lipschitz constraint in WGANs to improve performance.
problem WGANs do not always outperform other GAN variants due to imperfect implementation of the Lipschitz condition.
method Proposes a new dual form of Wasserstein distance (Sobolev duality) that relaxes the Lipschitz constraint but maintains gradient property.
result SWGAN, based on Sobolev duality, outperforms existing methods in experiments.
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.
Proposes a variational NNCC formulation for infinite dimensions.
problem Optimization and gradient flows in infinite-dimensional settings.
method Variational formulation of NNCC on c-convex domains.
result Wasserstein spaces inherit NNCC from their base space.
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.
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.
Entropy convexity characterizes strong energy condition in spacetimes.
problem Characterizing strong energy condition in nonsmooth spacetimes.
method Lifting fractional powers of Lorentz distance to probability measures and showing geodesic convexity of Boltzmann-Shannon entropy.
result Strong energy condition is equivalent to geodesic convexity of Boltzmann-Shannon entropy.
Efficiently simulates and calibrates the rough Bergomi model using Wasserstein distance.
problem High computational complexity in pricing and calibration of the rough Bergomi model.
method Developed a modified-sum-of-exponentials Monte Carlo scheme and a calibration approach based on Wasserstein-1 distance.
result The method achieves high pricing accuracy and improved parameter recovery, optimization stability, and out-of-sample performance.
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.
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-convex functions and (weak) Kantorovich potentials do not extend to the Lorentzian setting, but under suitable assumptions, they are locally semconvex. 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.
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.
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.