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.
Develops optimal transport in Lorentzian spaces with synthetic curvature bounds.
problem Synthetic curvature bounds for Lorentzian spaces.
method Optimal transport, convexity analysis of entropy functionals.
result Synthetic notion of timelike Ricci curvature lower bounds.
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. Study optimal transport on null hypersurfaces and null energy condition.
problem Optimal transport degeneracy on null hypersurfaces.
method Developed tools to characterize null energy condition using convexity properties of entropy.
result Optimal transport characterization of null energy condition.
Sharp inequality for Lorentzian spaces with timelike Ricci bounds.
problem Establishing bounds on achronal hypersurfaces in Lorentzian spaces.
method Optimal transport and synthetic TCDpe(K,N) spaces. result Sharp isoperimetric-type inequality for Lorentzian spaces.
Clarifies definitions of global hyperbolicity in various spaces.
problem Clarifying terminology in recent literature on global hyperbolicity.
method Comparing definitions in Lorentzian length spaces, optimal transport, and topological preordered spaces.
result The causal relation is a closed order and preserves compactness in all cases.
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…
New curvature measure for causal sets derived from optimal transport.
problem Capturing Ricci curvature in causal sets.
method Using Lorentzian optimal transport, novel curvature defined along maximal chains.
result Recovery of timelike Ricci curvature from order-theoretic data.
Stability of timelike Ricci bounds in low-regularity spacetimes.
problem Stability of synthetic timelike Ricci curvature bounds under C0-limits. method Constructing smooth approximations and analyzing limiting behavior via Lorentzian optimal transport.
result Impulsive gravitational waves satisfy synthetic timelike Ricci curvature lower bounds.
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.
Study timelike Ricci curvature bounds via optimal transport with Orlicz-type costs.
problem Characterize timelike Ricci curvature bounds.
method Optimal transport with Orlicz-type costs, convexity of relative entropy.
result Characterize timelike Ricci curvature lower bounds via convexity of relative entropy.
The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal tran…
Establishes a Lorentzian Lasry-Lions regularization theorem for functions on globally hyperbolic spacetimes.
problem Optimal transport with C1,1 regularizing pairs method Local semiconcavity and future-directed timelike superdifferentials
result Derives C1,1 regularizing pairs for optimal transport under general assumptions This survey introduces synthetic timelike Ricci curvature bounds in Lorentzian spaces.
problem Synthetic timelike Ricci curvature bounds in non-smooth Lorentzian spaces.
method Optimal transport and entropy tools.
result Synthetic version of Hawking's singularity theorem and synthetic characterisation of Einstein's vacuum equations.
The study proves timelike Ricci bounds for low regularity spacetimes using optimal transport.
problem Proving timelike Ricci bounds for spacetimes with low regularity.
method Using optimal transport to prove timelike measure-contraction property.
result Timelike curvature-dimension condition holds for C1,1 metrics. Synthetic framework for null hypersurfaces in non-smooth spacetimes.
problem Analyzing null hypersurfaces in non-smooth spacetimes.
method Develops synthetic null hypersurfaces using optimal transport and Lorentzian geometry.
result Synthetic null energy condition stabilizes under convergence and applies to low-regularity spacetimes.
Researchers found the longest arcs for specific sub-Lorentzian structures.
problem Finding the longest arcs for sub-Lorentzian structures.
method Optimal control problem with unbounded control set and concave cost functional. Sufficient conditions for existence of longest arcs proposed.
result Existence of the longest arcs for left-invariant three-dimensional contact sub-Lorentzian structures proved.
Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.
problem Understanding extremal trajectories and reachable set on anti-de Sitter plane.
method Geometric control theory and differential geometry.
result Construction of optimal synthesis and description of Lorentzian distance.
Two flat sub-Lorentzian problems on Martinet distribution differ in attainable set intersections.
problem Flat sub-Lorentzian structures on Martinet distribution.
method Analysis of attainable sets, optimal trajectories, sub-Lorentzian distances and spheres.
result The attainable set for the first problem intersects with the Martinet plane, while for the second it does not.
Study of a series of Lorentzian structures on SL(2,R) with SO(1,1) symmetry.
problem Global optimality of extremal trajectories in a series of Lorentzian structures.
method Analysis of a one-parametric series of left-invariant Lorentzian structures on SL(2,R) with SO(1,1) symmetry.
result Properties of the Lorentzian structures deform to those of the sub-Lorentzian structure in a limit case.
Study on Heisenberg group's Lorentzian problems using Pontryagin's principle.
problem Lorentzian problems on the Heisenberg group.
method Applied Pontryagin's maximum principle to obtain extremal trajectories.
result Parameterization of abnormal and normal extremal trajectories, investigation of reachability sets and existence of optimal trajectories.
New methods estimate transport-growth pairs in unbalanced optimal transport.
problem Statistical guarantees for Monge-type estimation in unbalanced optimal transport remain limited.
method Developed two estimators for transport-growth pairs under different setups.
result Achieved minimax optimal rate for estimation of transport-growth pairs.
Introduces statistical optimal transport for probabilistic lectures.
problem No specific problem stated; focuses on introduction.
method Lecture-based introduction to statistical optimal transport.
result Provides an introduction to statistical optimal transport.
NOT learns optimal transport plans, kernel costs improve performance.
problem NOT algorithm learns non-optimal plans with weak quadratic costs.
method Introduced kernel weak quadratic costs to improve NOT's performance.
result Kernel costs provide improved theoretical and practical guarantees.
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).
We show that every closed Lorentzian surface contains at least two closed geodesics. Explicit examples show the optimality of this claim. Refining this result we relate the least number of closed geodesics to the causal structure of the surface and the homotopy type of the Lorentzian metric.
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.
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.
In this paper, we present a novel and principled approach to learn the optimal transport between two distributions, from samples. Guided by the optimal transport theory, we learn the optimal Kantorovich potential which induces the optimal transport map. This involves learning two convex functions, by solving a novel mi…
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.
Review of modern computational optimal transport methods for biomedical applications.
problem Efficient computation of optimal transport for big data.
method Regularization-based and projection-based computational methods.
result Advancements in computational optimal transport methods for biomedical research.
Paper investigates optimal transport map estimation in infinite-dimensional spaces.
problem Estimating optimal transport maps in infinite-dimensional spaces is challenging.
method Characterizes γ-smoothness for optimal transport maps and develops a polynomial-rate estimator. result Shows polynomial-order minimax risk for optimal transport map estimation.
Study shows how optimal transport behaves in higher dimensions.
problem Characterizing optimal transport in higher dimensions with Euclidean distance.
method Investigates the small regularization limit of entropic optimal transport.
result The limiting transport plan is supported on transport rays and uniquely minimizes a relative entropy functional.
Study of longest arcs and cut loci in deformed anti de-Sitter spaces.
problem Existence and properties of time-like cycles in deformed Lorentzian manifolds.
method Analysis of universal covering, admissible curves, and Lorentzian geodesics.
result Identification of cut time and cut locus in deformed anti de-Sitter spaces.
A new model corrects inhomogeneity in Optimal Transport with Boundary.
problem Inhomogeneity in UROT models for Optimal Transport with Boundary.
method Proposed a modified entropic regularization term to make UROT models homogeneous.
result Homogeneous UROT model preserves properties of standard UROT while correcting inhomogeneity.
We introduce a version of Aubry-Mather theory for the length functional of causal curves in compact Lorentzian manifolds. Results include the existence of maximal invariant measures, calibrations and calibrated curves. We prove two versions of the Mather's graph theorem. A class of examples, the Lorentzian Hedlund exam…
Optimal transport as a loss for machine learning optimization problems has recently gained a lot of attention. Building upon recent advances in computational optimal transport, we develop an optimal transport non-negative matrix factorization (NMF) algorithm for supervised speech blind source separation (BSS). Optimal …
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.
Introduces a new divergence measure for optimal transport.
problem Optimal transport distances and information divergences.
method Infimal convolution formulation of proximal optimal transport divergence.
result Establishes connections to dynamic formulations and partial differential equations.
Optimal Transport enhances machine learning with new methods.
problem Comparing and manipulating probability distributions in machine learning.
method Probabilistic framework rooted in rich history and theory.
result New solutions in generative modeling and transfer learning.
Study optimal transport on simplex boundary, proving transport map and potential regularity.
problem Regularity of transport map and potential on simplex boundary.
method Boundary regularity results for optimal transport maps, exploiting simplex symmetries.
result Regularity properties of transport map and its convex potential.
Alternative proof of Michael-Simon-Sobolev inequality using optimal transport.
problem Proving the Michael-Simon-Sobolev inequality for submanifolds of codimension 2.
method Optimal transport techniques.
result Sharpness of the inequality for submanifolds of codimension 2.
Survey of linking information geometry and optimal transport.
problem Connecting two geometric frameworks for probability measures.
method Exploration of interactions and links between information geometry and optimal transport.
result Outstanding questions for both disciplines.
Bayesian approach to optimal transport with stochastic costs.
problem Inferring optimal transport plans with uncertain costs.
method Bayesian framework and Hamiltonian Monte Carlo (HMC) sampling.
result Inference of optimal transport plans under stochastic cost functions.
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…
Paper introduces a neural network for consistent estimation of optimal transport maps.
problem Statistically consistent estimation of optimal transport maps between probability distributions.
method Lipschitz-constrained GAN penalized by quadratic transportation cost.
result The generator converges uniformly to the optimal transport map as sample size increases.
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…
Optimal transport framework for zero-shot learning.
problem Generalized zero-shot learning of unseen classes.
method Conditional generative model and optimal transport between generated and real features.
result Optimal transport-based method outperforms state-of-the-art methods.