Paper relaxes optimal transport using convex functions for data science.
problem Optimal transport problem on finite spaces.
method Relaxation via strictly convex functions (Kullback-Leibler divergence, Bregman divergences). Gradient descent iterative process.
result Mathematical foundations and iterative process for the relaxed optimal transport problem.
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.
Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans π1,π2,… converges weakly to a transport plan π, then π is also optimal (between its marginals). Alfonsi, Corbetta and Jourdain asked whether the same property is true for th…
We explore the use of deep learning and deep reinforcement learning for optimization problems in transportation. Many transportation system analysis tasks are formulated as an optimization problem - such as optimal control problems in intelligent transportation systems and long term urban planning. Often transportation…
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).
A novel Federated Learning scheme using Optimal Transport for personalized model training.
problem Training models with data from clients having non-identically distributed data.
method Personalized Federated Learning scheme based on Optimal Transport (FedOT).
result FedOT scheme effectively transfers data from multiple distributions to a common domain and optimizes the prediction model.
Extends Optimal Transport to multiple agents, aiming for equitable and optimal distribution.
problem Sharing costs or goods equitably among multiple agents with different preferences.
method Minimizes the maximum transportation cost or maximizes the minimum utility.
result Provides a new algorithm faster than standard linear programming.
Study investigates duality and dual optimizers for various transport problems.
problem Existence and characterization of dual optimizers for adapted transport problems.
method Minimal assumptions, including causal and bicausal settings, are considered.
result No-arbitrage assumption leads to multicausal couplings and equivalent robust superhedging price computation.
New algorithm improves on existing methods for solving transport problems.
problem Finding a map to transport one distribution to another.
method Iterative Markovian Fitting (IMF) and Diffusion Schrödinger Bridge Matching (DSBM).
result DSBM significantly improves over previous SB numerics and recovers various transport methods.
Optimal transportation distances are a fundamental family of parameterized distances for histograms. Despite their appealing theoretical properties, excellent performance in retrieval tasks and intuitive formulation, their computation involves the resolution of a linear program whose cost is prohibitive whenever the hi…
Unified deep learning framework solves various optimal transport problems.
problem Solving variational problems in optimal transport with computational challenges.
method Unified deep learning framework leveraging dual formulation of Lagrangians.
result Outperforms previous approaches in single-cell trajectory inference.
Paper introduces information-constrained optimal transport, generalizing Talagrand's inequality.
problem Optimal transport problem with information constraints.
method Information constrained variation of optimal transport, using Marton's approach.
result Recovery of concentration of measure results and solution to Cover's open problem.
Proves hardness of semi-discrete optimal transport and proposes regularization methods.
problem Computing Wasserstein distance between discrete and non-discrete probability measures.
method Proves hardness, introduces distributionally robust dual optimal transport, regularizes primal objective, uses stochastic gradient descent.
result Regularization schemes and improved convergence guarantees for semi-discrete optimal transport problems.
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. Optimal Transport (OT) naturally arises in many machine learning applications, yet the heavy computational burden limits its wide-spread uses. To address the scalability issue, we propose an implicit generative learning-based framework called SPOT (Scalable Push-forward of Optimal Transport). Specifically, we approxima…
The problem for consistency between linear transports along paths and real bundle metrics in real vector bundles is stated. Necessary and/or sufficient conditions, as well as conditions for existence, for such consistency are derived. All metrics (resp. transports) consistent with a given transport (resp. metric) are e…
Study uses weak transport for non-convex costs in fixed-income markets.
problem Characterizing optimal caplet pricing in fixed-income markets.
method Introduced weak optimal transport for non-convex costs, reduced general costs to convex problems.
result Established robust super-replication results for fixed-income markets.
Study optimal transport for robust optimization, showing how adversary's strategy relates to regularization.
problem Optimizing under uncertain parameters with a fictitious adversary reshaping a reference distribution.
method Introduces optimal transport and regularization to relate robustification to variation and Lipschitz norms.
result Conditions for existence and computability of Nash equilibrium between decision-maker and adversary.
A new method for efficient optimal partial transport in 1D.
problem Limitation of equal mass assumption in optimal transport.
method Sliced Optimal Partial Transport (Sliced-OPT) algorithm.
result Sliced-OPT demonstrates computational and accuracy benefits.
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…
Optimal transport reformulates multiple quantile hedging problem.
problem Multiple quantile hedging problem in incomplete markets.
method Reformulated as Monge optimal transport problem, introduced Kantorovitch version, proved no duality gap.
result Multiple quantile hedging problem can be seen as semi-discrete optimal transport problem.
New algorithm detects anomalies by forcing samples to displace mass in low-density regions.
problem Detecting anomalies in datasets.
method Mass Repulsing Optimal Transport (MROT) approach.
result Our algorithm improves anomaly detection over existing methods.
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.
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.
This paper presents a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a finite dimensional one which corresponds to optimizing a neural network wi…
Wasserstein GANs with Gradient Penalty compute a different optimal transport problem called congested transport.
problem Training generative models to produce high-quality synthetic data.
method Wasserstein GANs with Gradient Penalty (WGAN-GP) approach to calculate the Wasserstein 1 distance.
result WGAN-GP computes the minimum of the congested transport problem, not the Wasserstein 1 distance.
New curvature measure for optimal transport with specific cost function.
problem Optimal transport with specific cost function.
method Proposed generalized curvature measure.
result Non-negativity of the generalized curvature implies displacement convexity.
Develops theory for conditional optimal transport in infinite-dimensional spaces.
problem Bayesian inference with functional parameters in infinite-dimensional spaces.
method Theory of constrained optimal transport for block-triangular maps.
result Regularity estimates on conditioning maps from prior to posterior.
Researchers analyze inverse optimal transport, deriving theoretical and empirical insights.
problem Understanding the inverse problem of inferring cost matrices from optimal couplings.
method Formalized and analyzed using entropy-regularized optimal transport, with theoretical and empirical contributions.
result Characterization of the manifold of cross-ratio equivalent costs and derivation of an MCMC sampler.
New toolkit for directed distances improves flexibility of OT problems.
problem Optimal transport problems with constraints.
method Directed distances between quantile functions.
result Flexibility in solving OT problems enhanced.
Develops a new duality between entropy martingale optimal transport and nonlinear pricing-hedging.
problem Entropy Martingale Optimal Transport problem and its associated optimization problem.
method Combines Entropy Optimal Transport and Martingale Optimal Transport theories, with novel penalization terms and constraints.
result Establishes a nonlinear robust pricing-hedging duality, covering various known robust results.
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.
Study MinMax methods for optimization problems, including optimal transport.
problem Optimization problems, especially optimal transport.
method MinMax framework, regularization, neural networks, approximation theorems.
result Justification of neural networks for solving optimization problems.
A JAX toolbox solves optimal transport problems for point clouds and histograms.
problem Optimal transport problems between point clouds and histograms.
method Automatic and custom reverse mode differentiation, vectorization, just-in-time compilation, and accelerators support.
result Solves a wide range of optimal transport problems including regularized OT, barycenters, Gromov-Wasserstein, and low-rank solvers.
This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Sev…
Researchers developed a new Riemannian manifold for SPD matrix-valued optimal transport problems.
problem Optimal transport between SPD matrix-valued measures.
method Formulated as a generalized optimal transport problem with block SPD matrices, endowed with a novel Riemannian manifold structure.
result The novel Riemannian manifold allows solving SPD matrix-valued optimal transport problems using Riemannian optimization.
Estimates discontinuous optimal transport maps between a discrete and continuous distribution.
problem Estimating discontinuous optimal transport maps between a discrete and continuous distribution.
method Entropic optimal transport estimator, computationally efficient.
result The estimator converges at the minimax-optimal rate n−1/2 in the semi-discrete setting. New forms of multi-marginal POT problem derived for computational efficiency.
problem Optimizing transport between multiple unbalanced measures with limited supports.
method Developed two equivalence forms of the POT problem and an optimization algorithm, ApproxMPOT.
result ApproxMPOT algorithm achieves optimal value with complexity ildeO(m3(n+1)m/ε2). In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the discrete optimal transport, discrete Monge-Ampere equation and the power diagram in computa…
The article resolves complex structures in transport twistor spaces, proving a Newlander-Nirenberg theorem.
problem Degenerate complex structures in transport twistor spaces.
method Holomorphic blow-down structure maps to resolve degeneracy and gain insight into complex geometry.
result Global and local β-maps for various metrics, proving a Newlander-Nirenberg theorem for degenerate complex structures.
Consider a multiperiod optimal transport problem where distributions μ0,…,μn are prescribed and a transport corresponds to a scalar martingale X with marginals Xt∼μt. We introduce particular couplings called left-monotone transports; they are characterized equivalently by a no-crossing property…
We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Ampère type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We …
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.
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…
A new method for optimal transport using neural ODEs that preserves marginal constraints.
problem Optimal transport between two continuous distributions with specific cost functions.
method Iterative construction of neural ODEs to minimize transport cost while preserving marginal constraints.
result Monotonic interior approach that decreases transport cost efficiently.
The paper studies optimal transport in linear quadratic systems and derives interpolation inequalities.
problem Optimal transport problem in Linear Quadratic optimal control systems.
method Well-posedness of the Monge problem, regularity of optimal transport map, displacement interpolation of measures.
result Derivation of general interpolation inequalities for entropy functionals.
Study proves convergence of subgradients for optimal transport-based objectives.
problem Ensuring statistical consistency and optimization stability in transport-based models.
method Proves graphical convergence of subdifferentials to the subdifferential of the population objective.
result Standard subgradient methods consistently approach stationary points of the population-level problem.
Meta Optimal Transport learns from past problems to solve similar OT problems faster.
problem Solving similar optimal transport problems repeatedly from scratch is inefficient.
method Amortized optimization to predict optimal transport maps from past solutions.
result Meta OT models can solve new problems faster than standard methods.