Brenier isotonic regression extends multi-output isotonic regression using optimal transport.
problem Enforcing cyclic monotonicity in multi-output regression.
method Leverage Kantorovich's optimal transport to find cyclically monotone couplings.
result Brenier isotonic regression outperforms baselines in probability calibration.
Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is comparable to the distance between the original densities. Therefore, optimal transpo…
We analyze errors in filtering algorithms using optimal transport.
problem Estimation errors in optimal transport-based filtering algorithms.
method Systematic analysis of estimation errors for conditional Brenier maps.
result Demonstrates effectiveness and practical potential of the optimal transport filtering algorithm.
A new method reformulates Optimal Transport Conditional Flow Matching using proximal operators.
problem Optimal Transport Conditional Flow Matching (OT-CFM) for generating models.
method Reformulate OT-CFM using proximal operators and extended Brenier potential.
result OT-CFM dynamics are terminally normally hyperbolic for manifold-supported targets.
Neural network implementation of Brenier's polar factorization for vector fields.
problem Implementing Brenier's polar factorization theorem for vector fields using neural networks.
method Parameterizing the convex function u as an input convex neural network and estimating the measure-preserving map M. result Practical neural implementation of Brenier's polar factorization theorem.
The paper improves OT map estimation rates without strict assumptions.
problem Estimating optimal transport maps under practical conditions.
method Developed new convergence rates and scalable algorithms.
result Improved convergence rates for OT map estimation without restrictive assumptions.
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…
A new discrete formula connects vertex and edge distributions on graphs.
problem Optimal transport on graphs with mixed vertex and edge distributions.
method Discrete transport equation and Benamou-Brenier formulation.
result Classification of all Wasserstein-1 geodesics on graphs.
New method finds closest martingale to Brownian motion.
problem Finding optimal martingale interpolating marginals.
method Martingale Sinkhorn algorithm, iterative scheme.
result Algorithm yields Bass potential in arbitrary dimension.
Estimates conditional Brenier maps using entropic optimal transport.
problem Non-parametric estimation of conditional Brenier maps.
method Entropic optimal transport for scalable non-parametric estimation.
result Entropic optimal transport maps asymptotically converge to conditional Brenier maps.
Extends optimal transport to dynamic and martingale settings.
problem Dynamic and martingale relaxation of optimal transport problems.
method Extends Benamou-Brenier formula to weak optimal transport and introduces barycentric optimal transport.
result Relates barycentric optimal transport to martingale Benamou-Brenier formula.
This work builds the connection between the regularity theory of optimal transportation map, Monge-Ampère equation and GANs, which gives a theoretic understanding of the major drawbacks of GANs: convergence difficulty and mode collapse. According to the regularity theory of Monge-Ampère equation, if the support of the …
Modified Wasserstein metric for Gaussian distributions, invariant to isometries.
problem Distance measurement for latent Gaussian distributions invariant to isometries.
method Modified Benamou-Brenier approach leading to a Procrustes Wasserstein metric.
result For Gaussian distributions, the metric reduces to Euclidean distance between eigenvalues.
New framework uses PDE for no-regret generative modeling.
problem Developing efficient generative models for complex distributions.
method Iterative refinement of Brenier maps using mirror gradient descent.
result Converges to optimal Brenier map under various step-size schedules.
The paper constructs denoisers that recover the Brenier map from higher-order score functions.
problem Estimating the Brenier map from noisy data.
method Constructs a hierarchy of denoisers using higher-order score functions.
result The T∞ denoiser recovers the Brenier map from the additive Gaussian model. Optimal transport explored on a specific geometric space.
problem Optimal transport problem in sub-Lorentzian Heisenberg group.
method Synthetic metric spacetime structure analysis and sub-Lorentzian version of Brenier's theorem.
result Established sub-Lorentzian version of Brenier's theorem and derived Monge-Ampère equation.
Geometric Bass martingales linked to Brownian motion and geometric Brownian motion.
problem Modeling continuous martingales with prescribed initial and terminal distributions.
method Developed geometric Bass martingales and established their properties.
result Explicit bijection and representation of geometric Bass martingales.
A new diffusion method approximates Schrödinger bridge with improved convergence.
problem Approximating Schrödinger bridge with Langevin diffusion.
method Leveraging Langevin diffusion to approximate Schrödinger bridge.
result The difference between the two approximations is proportional to the score function.
New bounds on optimal transport regularization show faster convergence rates than previously known.
problem Understanding the localization rate of Quadratically Regularized Optimal Transport (QOT) optimizers.
method Established lower bounds and derived mean-squared deviation controls for QOT optimizers.
result Lower bound of support concentration rate εd+21 in directed Hausdorff distance. Researchers found counterexamples to conjectures about optimal transport maps on curved spaces.
problem Extending Caffarelli's contraction theorem to curved spaces.
method Constructing counterexamples to precise conjectures.
result Found counterexamples to Milman's conjectures about optimal transport maps on curved spaces.
In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. We suggest a Benamou-Brenier type formulation of the martingale transport proble…
Paper solves DRO for continuous distributions with iterative algorithms.
problem Distributionally robust optimization with continuous worst-case distributions.
method Iterative algorithm for global convergence, leveraging Brenier's theorem and JKO scheme.
result Achieves global convergence under mild assumptions for minimax problems.
We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form of the optimal transportation problem, the Brenier theorem states that the optim…
By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present…
These notes briefly summarize the lectures for the Summer School "Optimal transportation: Theory and applications" held by the second author in Grenoble during the week of June 22-26, 2009. Their goal is to describe some recent results on Brenier's variational models for incompressible Euler equation.
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…
Correctly estimating the discrepancy between two data distributions has always been an important task in Machine Learning. Recently, Cuturi proposed the Sinkhorn distance which makes use of an approximate Optimal Transport cost between two distributions as a distance to describe distribution discrepancy. Although it ha…
Improved tracking of tangled point sources using Riemannian metrics.
problem Tangled point source trajectories in temporal stacks.
method Lifting to higher-dimensional space of roto-translation group, new regularisation based on relaxed Reeds-Shepp metric.
result Reconstruction and untangling of trajectories even from numerical standpoint.
New model generates data without mode collapse or mixture.
problem Mode collapse and mode mixture in generative models.
method Inspired by AE-OT, improved sample generation algorithm.
result Free from mode collapse and mixture issues.
A new method steers Gaussian distributions with minimal effort.
problem Steering high-dimensional Gaussian distributions efficiently.
method Sliced feedback controller using one-dimensional projections and averaging.
result The method steers Gaussian distributions to targets efficiently.
New theorem shows nearly spherical manifolds can be mapped from spheres.
problem Generalizing Caffarelli's theorem to nearly spherical manifolds.
method Optimal transport map on the sphere, stability result.
result Every nearly spherical manifold can be mapped from a sphere.
Variational autoencoders often collapse, showing latent variables are non-identifiable.
problem Posterior collapse in variational autoencoders due to non-identifiable latent variables.
method Proves latent variable non-identifiability causes posterior collapse. Proposes latent-identifiable models using Brenier maps and input convex neural networks.
result Latent-identifiable models resolve posterior collapse and provide meaningful representations.
The paper optimizes estimating transport maps between distributions.
problem Estimating optimal transport maps between distributions.
method Plugin approach using optimal couplings and extensions.
result Minimax optimality of the proposed estimators.
Let K be an irreducible and reversible Markov kernel on a finite set X. We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wassers…
Novel stability bounds for OT maps improve density estimation.
problem Estimating optimal transport maps between probability distributions.
method Developed novel stability bounds for OT maps, reducing the problem to density estimation.
result Stability bounds allow for sharper guarantees without smoothness assumptions.
We show in this note that the Sobolev Discrepancy introduced in Mroueh et al in the context of generative adversarial networks, is actually the weighted negative Sobolev norm ∣∣.∣∣H˙−1(νq), that is known to linearize the Wasserstein W2 distance and plays a fundamental role in the dynamic formulation of…
New optimal transport method handles mass creation and destruction.
problem Optimal re-balancing of portfolios with mass creation or destruction.
method Formalizes an optimal transport problem with mass-change factor.
result Existence of optimal transport plans and maps established.
In this paper we revisit the anisotropic isoperimetric and the Brunn-Minkowski inequalities for convex sets. The best known constant C(n)=Cn7 depending on the space dimension n in both inequalities is due to Segal [\ref{bib:Seg.}]. We improve that constant to Cn6 for convex sets and to Cn5 for centrally sy…
A new method to estimate optimal transport maps without constraints.
problem Challenges in fitting optimal transport maps with neural networks.
method Introducing a Monge gap regularizer to estimate OT maps without architectural constraints.
result The proposed method significantly outperforms other baselines in practice.
Large Sinkhorn couplings improve flow models in data generation tasks.
problem Training flow models with optimal transport couplings.
method Using large batches of source and target points, and applying entropic regularization with a low ε. result Flow models perform better with large Sinkhorn couplings and low regularization.
We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiati…
Constructs supermartingale couplings with full marginals constraints.
problem Optimal transport for supermartingale couplings with multiple marginals.
method Markovian iteration of one-period optimal supermartingale couplings.
result Explicit construction of supermartingale processes solving optimal transport problem.
Paper introduces ICGNs to model convex gradients.
problem Modeling convex gradients efficiently.
method Integrates Jacobian-vector product in a neural network.
result Single layer ICGN outperforms single layer ICNN in fitting.
A mesh-free method solves continuum-marginal optimal transport problems.
problem Recovering minimum-energy velocity fields from time-continuous probability marginals.
method Embeds weak continuity equation in a reproducing kernel Hilbert space, optimizing with mini-batch stochastic methods.
result Accurately recovers drift and maintains marginal consistency in synthetic experiments.
This work introduces methods to compute optimal Monge maps and learn elastic costs for efficient data mapping.
problem Efficiently mapping one probability distribution to another using elastic costs.
method Proposes numerical methods to compute optimal Monge maps and a learning loss for parameterized regularizers.
result Proves the optimality of computed Monge maps and learns the parameters of elastic costs.
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.
Paper uses optimal transport for Bayesian filtering, deriving new EnKF and FPF formulations.
problem Bayesian filtering for nonlinear systems with non-Gaussian observations.
method Optimal transport theory applied to Bayes' law, constructing Brenier maps.
result New variational formulations of EnKF and FPF for non-Gaussian settings.
Generative sampler learns velocity fields for efficient posterior inference.
problem Sampling from complex posterior distributions in high dimensions.
method Generative multivariate posterior sampler via flow matching, learning a velocity field for a deterministic transport map.
result Conditional Brenier map enables fast generation of credible sets with theoretical consistency guarantees.