Paper derives and applies a parallel transport equation on Lie groups.
problem Efficiently solving parallel transport on Lie groups with left-invariant metrics.
method Derives a parallel transport equation in Lie algebra, applies it to SE(3), and compares to existing methods.
result Stable and efficient parallel transport implementation on Lie groups.
New metric for probability measures connects physics and geometry.
problem Developing a new metric for probability measures.
method Transport Hessian metric, formulated dynamical systems.
result Connections to physics equations and mathematical models.
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…
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.
Noise stabilizes solutions to transport equations, preventing blow-up.
problem Proving global existence and uniqueness of solutions to stochastic transport equations.
method Characteristics-based techniques exploiting the geometric structure of transport equations.
result Noise prevents blow-up in deterministic solutions and ensures global existence and uniqueness of solutions.
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.
Exposes how Hessian manifold duality aids in solving optimal transport problems.
problem Solving Monge-Ampère equations and understanding mirror symmetry.
method Explains duality theory for Hessian manifolds and its application to optimal transport.
result Provides a natural setting for optimal transport and solves Monge-Ampère equations.
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.
This paper uses normalizing flows to approximate transport maps between densities.
problem Approximating transport maps between given densities.
method Construct time-dependent controls using normalizing flows.
result Provides bounds on the number of switches for piecewise constant approximations.
Unified framework for constructing kernels for transport equations and Koopman eigenfunctions.
problem Constructing kernels for transport equations and Koopman eigenfunctions.
method Three methods: variational principle, Green's function, and resolvent operator.
result Kernels constructed via these methods are identical under mild assumptions.
The goal of the paper is to give an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the …
Steerable neural ODEs on homogeneous spaces for equivariant feature dynamics.
problem Learning continuous-time equivariant dynamics of vector-valued features on homogeneous spaces.
method Introduces steerable neural ordinary differential equations on homogeneous spaces, interpreting features as sections of associated vector bundles over M. result Steerable NODEs are G-equivariant when the flow and connection are G-invariant, and they incorporate existing models. Survey of Sinkhorn algorithm for optimal transport, emphasizing its geometric origins.
problem Solving optimal transport problems efficiently and accurately.
method Discretization of a non-linear integral equation.
result Geometric interpretation and discretization of the Sinkhorn algorithm.
The displacement and deviation vectors in spaces (manifolds), the tangent bundle of which is endowed with a transport along paths, are introduced. In case these spaces are equipped with a linear connection, the deviation equations (between arbitrary, geodesic or not, paths) in such spaces are investigated.
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.
We call a manifold with torsion and nonmetricity the metric-affine manifold. The nonmetricity leads to a difference between the auto parallel line and the extreme line, and to a change in the expression of the Frenet transport and moving basis. The torsion leads to a change in the Killing equation. We also need to add …
We study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher's information metric on the space of smooth probability densities. The right reduced geodesic equations are higher-dimensional generalisations of the μ--H…
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 …
Study mass transport in low-diffusivity using Lagrangian coordinates.
problem Mass preserving transport of passive tracers in low-diffusivity limit.
method Lagrangian coordinates, time-averaged diffusion equation, weighted manifold structure.
result Leading order asymptotics extend to dominant nontrivial singular value in low-diffusivity limit.
A new algorithm for parallel transport on shape spaces is presented and compared to existing methods.
problem Statistical analysis of shape data, especially in time series and optimization.
method Pole ladder algorithm for parallel transport on Kendall shape spaces, compared to integration methods.
result The pole ladder algorithm is a more efficient method for parallel transport.
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.
We study a parabolic equation for finding solutions to the optimal transport problem on compact Riemannian manifolds with general cost functions. We show that if the cost satisfies the strong MTW condition and the stay-away singularity property, then the solution to the parabolic flow with any appropriate initial condi…
New proof of Schwarzschild stability using geometric gauge.
problem Linear stability of Schwarzschild spacetime under gravitational perturbations.
method Employing a new geometric gauge and exploiting the structure of transport equations.
result Established both orbital and asymptotic stability for linearised quantities.
Geometric framework for Newton's equations on diffeomorphism groups.
problem Modeling fluid dynamics and related systems on geometric spaces.
method Geodesic approach and infinite-dimensional information geometry.
result Unified framework for various fluid dynamics equations.
Paper proposes a new approach to optimal transport for vector and matrix densities.
problem Optimal transport for vector and matrix densities with positivity and action transitivity constraints.
method Gauge-theoretic approach using semi-direct product groups of diffeomorphisms and gauge transformations.
result Bures-type metrics on semi-direct product groups relate to Wasserstein-type metrics on vector and matrix densities via Riemannian submersions.
This work clarifies different transport map constructions and their causal interpretations.
problem Identifying distinct transport map constructions and their equivalence.
method Comparative analysis of three transport map constructions: cyclically monotone, quantile-preserving, and triangular monotone.
result Conditions for equivalence of different transport map constructions.
New control methods improve dynamic measure transport paths.
problem Improving paths for dynamic measure transport.
method Connecting mean-field games to optimization problems for learning paths, advocating for smoothness of velocities.
result Our method recovers more efficient and smooth transport models compared to untilted paths.
The paper examines soliton surfaces using a parallel transport frame field in 4D space.
problem Geometric properties of soliton surfaces associated with the Betchov-Da Rios equation.
method Parallel transport frame field approach in four-dimensional Euclidean space.
result Characterization of soliton surfaces as flat, minimal, semi-umbilic, or Wintgen ideal.
The group of diffeomorphisms of a compact manifold endowed with the L^2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon …
Local well-posedness proved for Bartnik static extension near Schwarzschild spheres.
problem Proving well-posedness for the Bartnik static extension problem near Schwarzschild spheres.
method Introduced a geodesic gauge to formulate governing equations as coupled elliptic and transport equations; used Bochner-measurable functions for transport equations.
result Established local well-posedness for arbitrary Bartnik data near Schwarzschild spheres, including those with small mean curvature.
New framework for gravitational perturbations of Kerr spacetimes, focusing on stability.
problem Stability of Kerr spacetimes to gravitational perturbations.
method New geometric framework with tailored null frames and gauge, reformulating Einstein equations.
result Derivation of linearised vacuum Einstein equations in the new framework.
The paper calibrates SPX and VIX options using optimal transport.
problem Joint calibration of SPX and VIX options or futures.
method Semimartingale optimal transport problem with PDE formulation and dual formulation.
result The model accurately calibrates SPX, VIX options, and futures simultaneously.
Discontinuous Finite Element Methods (DFEM) have been widely used for solving Sn radiation transport problems in participative and non-participative media. In the DFEM Sn methodology, the transport equation is discretized into a set of algebraic equations that have to be solved for each spatial cell and angular d…
In a coordinate free form are found the (deviation) equations satisfied by the (infinitesimal) deviation vector, relative velocity, relative momentum, relative acceleration and relative energy of two point particles in a differentiable manifold the tangent bundle of which is endowed with a linear transport along paths,…
Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.
problem Connection between Yang-Mills fields and modified Lévy Laplacians on 4-manifolds.
method Analysis of modified Lévy Laplacians and their relation to Yang-Mills equations under nontrivial holonomy groups.
result Existence of a modified Lévy Laplacian related to Yang-Mills self-duality equations.
Study of a risk-averse informed trader in a multi-asset market with non-Gaussian prices.
problem Existence of equilibrium in a multi-asset market with non-Gaussian prices and a risk-averse informed trader.
method Constructed equilibrium using Fokker-Planck equation and coupled partial differential equations with an optimal transport constraint.
result Equilibrium exists in a market with multiple assets and non-Gaussian prices.
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 algorithm solves online resource allocation problems efficiently.
problem Dynamic resource allocation in operations research.
method Minimal Selection Principle and MSoE algorithm.
result Ensures optimal cumulative regret bounds in dynamic resource allocation.
This paper refines the Gaussian Sinkhorn algorithm for general multivariate models.
problem Finite-dimensional solutions for general Gaussian multivariate models.
method Recursive formulation of the Sinkhorn algorithm for Gaussian models, including closed form expressions of entropic transport maps and Schrödinger bridges.
result Refined convergence analysis of Gaussian Sinkhorn algorithms.
Solves a general class of free boundary Monge-Ampère equations.
problem Optimal transport with degenerate densities and geometric problems.
method Analyzes a specific class of Monge-Ampère equations and their applications.
result Solves the equations for a general class, including applications to optimal transport and geometric problems.
Gravity derived from thermodynamics via optimal transport.
problem Equivalence between gravity and thermodynamics.
method Linking optimal transport to Raychaudhuri equation in warped-product spacetimes.
result Equivalence between gravity and concavity of entropy under time evolution.
This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.
problem Approximating the Laplace-Beltrami operator using optimal transport with quadratic regularization.
method Deriving first-order optimal potentials and analyzing the convergence of discrete Laplace operators.
result The discrete Laplace operators converge to the Laplace-Beltrami operator on smooth manifolds.
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…
We give a new probabilistic construction of solutions to real Monge-Ampère equations in R^n satisfying the second boundary value problem with respect to a given target convex body P) which fits naturally into the theory of optimal transport. More precisely, certain beta-deformed permanental (bosonic) N-particle point p…
Study non-Gaussian measures' concentration properties in metric spaces.
problem Concentration properties for non-linear Gaussian functionals with non-Gaussian tails.
method Prove generalised Transportation-Cost Inequalities (TCIs) for specific functionals.
result Extended TCIs for rough volatility and Parabolic Anderson Model.
On compact manifolds which are not simply connected, we prove the existence of "fake" solutions to the optimal transportion problem. These maps preserve volume and arise as the exponential of a closed 1 form, hence appear geometrically like optimal transport maps. The set of such solutions forms a manifold with dimensi…
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 …
In this paper we consider Monge-Ampère equations on compact Hessian manifolds, or equivalently Monge-Ampère equations on certain unbounded convex domains Ω⊆Rn, with a periodicity constraint given by the action of an affine group. In the case where the affine group action is volume-preserving, i.e.,…