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.
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.
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.
Study on regularity of optimal transport maps on convex domains with quadratic cost.
problem Regularity of optimal transport maps between convex domains with quadratic cost.
method Analysis of Cα-densities and C1,α boundary conditions, monotonicity formula for optimal transport maps. result Proves C1,1−ε-regularity for nondegenerate Cα-densities and C2,α-regularity for C1,α boundary. 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. 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.
New approach to sparse optimal transport for matching tokens with experts.
problem Sparse matching of tokens with experts in neural networks.
method Sparsity-constrained optimal transport with cardinality constraints.
result Solves nonconvex cardinality constraints with gradient methods.
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.
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…
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 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…
We propose an SDP relaxation for the Gromov-Wasserstein distance, providing globally optimal solutions.
problem Matching objects between incomparable spaces using the Gromov-Wasserstein distance.
method Semi-definite programming (SDP) relaxation of the GW distance.
result The SDP relaxation provides globally optimal solutions for the GW distance in some instances.
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.
We propose a novel end-to-end non-minimax algorithm for training optimal transport mappings for the quadratic cost (Wasserstein-2 distance). The algorithm uses input convex neural networks and a cycle-consistency regularization to approximate Wasserstein-2 distance. In contrast to popular entropic and quadratic regular…
We consider the entropic regularization of discretized optimal transport and propose to solve its optimality conditions via a logarithmic Newton iteration. We show a quadratic convergence rate and validate numerically that the method compares favorably with the more commonly used Sinkhorn--Knopp algorithm for small reg…
A new method matches measures across different spaces using cost-regularized optimal transport.
problem Matching measures in different spaces without aligned data.
method Cost-regularized optimal transport formulation to match measures across two Euclidean spaces.
result Demonstrated applicability to single-cell spatial transcriptomics/multiomics matching tasks.
The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov's curvature bounded below, Monge's transport problem for the quadratic cost admits a unique solution.
HyCNNs improve convex function learning and optimal transport.
problem Learning and optimizing convex functions efficiently.
method Combining Maxout networks and ICNNs to create a new neural architecture.
result HyCNNs require fewer parameters and outperform existing methods in convex tasks.
Pushing a little forward an approach proposed by Villani, we are going to prove that in the Riemannian setting the condition ∇2f<g implies that f is c-concave with respect to the quadratic cost as soon as it has a sufficiently small C1-norm. From this, we deduce a sufficient condition for the optimalit…
Adaptive, sparse graphs improve learning performance.
problem Inappropriate kNN for varying sampling density or noise.
method Quadratically regularised optimal transport.
result Graphs outperform kNN in learning applications.
This work broadens optimal transport map estimation theory to stochastic settings.
problem Existing theory for optimal transport map estimation is restricted to deterministic maps under specific conditions.
method Introduces a novel metric for evaluating stochastic maps, develops computationally efficient estimators with robust guarantees.
result First general-purpose theory for map estimation compatible with real-world stochastic applications.
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.
Neural solver computes Wasserstein geodesics and velocity fields efficiently.
problem Computing Wasserstein geodesics and velocity fields efficiently.
method Sample-based neural network approach to solve the minimax problem.
result Directly samples from target distribution and estimates velocity field.
New method AM learns optimal vector fields for entire distribution sequences, matching OT.
problem Optimal Transport (OT) problem in generative modeling.
method Action Matching (AM) method learns optimal vector fields for a sequence of distributions.
result AM method achieves optimal transport by learning vector fields for entire distribution sequences.
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.
Paper solves Gromov-Wasserstein for point clouds efficiently.
problem Quantifying similarity between two formations or shapes.
method Reformulates QAP as low-rank concave quadratic optimization problem.
result Global solution for large-scale problems with thousands of points.
Extends martingale Schrödinger bridge to arbitrary dimensions and characterizes it.
problem Tackles the martingale Schrödinger bridge in arbitrary dimensions.
method Identifies continuous-time counterpart and relates to variational problems.
result Continuous martingale Schrödinger bridge coincides with Föllmer martingale in irreducible case.
Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.
problem Proving sharp Sobolev inequalities on noncompact Riemannian manifolds with non-negative Ricci curvature.
method Using Optimal Mass Transportation with quadratic distance cost.
result Sharp Lp-Sobolev and Lp-logarithmic Sobolev inequalities established for p>1 and p=1. Proposes a novel approach for cluster-aware matching using Laplacian Optimal Transport.
problem Matching point clouds with intrinsic cluster structure requires robust region-to-region alignment over precise point-to-point correspondence.
method Laplacian Optimal Transport (LapOT) with regularization for cluster-aware matching and Refined Simultaneous Clustering (RSC) for consistent partitions.
result Laplacian Optimal Transport produces more consistent and meaningful alignments between point clouds.
New algorithms solve weak optimal transport problems for nonlinear costs.
problem Computing weak optimal transport with nonlinear costs.
method Mirror descent algorithms for primal and dual versions of WOT.
result Solutions for WOT and WOTUK compared with classical OT.
A new method for manifold learning using sparse regularised optimal transport.
problem Detecting latent manifolds in high-dimensional data with noisy observations.
method Proposes a symmetric version of optimal transport with quadratic regularisation to construct a sparse and adaptive affinity matrix.
result The method outperforms competing methods in numerical experiments and demonstrates robustness to heteroskedastic noise.
A new method learns straight trajectories in one step for optimal flow matching.
problem Learning flows with straight trajectories for fast inference.
method Optimal Flow Matching (OFM) approach using convex functions for vector fields.
result Recovering straight OT displacements in just one FM step for quadratic transport.
Optimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry originates from coordinate-invariant properties of statistical inference. Their con…
New stability bounds for Sinkhorn's algorithm in entropic optimal transport.
problem Stability and convergence of Sinkhorn's algorithm for entropic optimal transport.
method Semiconcavity approach to analyze stability and convergence.
result Exponential convergence of Sinkhorn's algorithm under semiconcavity conditions.
MF-PID uses interacting samples to efficiently transport probability mass.
problem Efficiently transporting probability mass in generative models.
method Introducing Mean-Field Path-Integral Diffusion (MF-PID) where samples become interacting agents.
result MF-PID achieves 19-24% reductions in control energy for demand-response control of energy systems.
LOT improves optimal transport for large datasets.
problem Efficient optimal transport for large datasets.
method Low-rank optimal transport (LOT) restricts search to low-nonnegative rank couplings.
result LOT complements and improves upon entropic regularization.
Scalable algorithm for computing Wasserstein-2 barycenters without bias.
problem Computing Wasserstein-2 barycenters efficiently and accurately.
method Input convex neural networks and cycle-consistency regularization.
result Our approach avoids introducing bias and does not require minimax optimization.
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…
New method learns disentangled representations using Gromov-Monge maps.
problem Learning disentangled representations from unlabelled data.
method Introduces a novel approach based on Gromov-Monge maps to preserve geometric features while aligning data distributions.
result Demonstrates effectiveness on four benchmarks, outperforming other methods.
New algorithm for low-rank optimal transport with improved interpretability and efficiency.
problem Quadratic scaling of optimal transport coupling matrix for massive datasets.
method Factor Relaxation with Latent Coupling (FRLC) algorithm.
result Superior performance on diverse applications including graph clustering and spatial transcriptomics.
Two log-linear approximations speed up optimal transport for deep learning applications.
problem Computing optimal transport in high dimensions is computationally expensive.
method Locality-sensitive hashing (LSH) and Nyström approximation with LSH-based sparse corrections.
result Log-linear time algorithms for entropy-regularized OT perform well in high-dimensional spaces.
A novel approach to computing barycenters on graph-supported probability measures.
problem Computing weighted averages of measures on graphs.
method Dynamic optimal transport formulation on the simplex, gradient descent on the probability simplex.
result Intrinsic gradient descent provides a coherent framework for synthesizing and analyzing measures on graphs.
Let (X,L) be a (semi-) polarized complex projective variety and T a real torus acting holomorphically on X with moment polytope P. Given a probability density g on P we introduce a new type of Monge-Ampere measure on X, defined for singular T-invariant metrics on the line bundle L, generalizing the ordinary Monge-Amper…
TreeDSB solves mOT problems on tree-structured costs for Wasserstein barycenters.
problem Optimal transport with multiple marginals and tree-structured quadratic costs.
method Tree-based Diffusion Schrödinger Bridge (TreeDSB) for continuous and dynamic solutions.
result TreeDSB efficiently computes Wasserstein barycenters in high dimensions.
The paper explores fair regression and classification under demographic parity constraints.
problem Ensuring fairness in regression and classification models under demographic parity constraints.
method Characterizes the optimal fair regression function using a barycenter problem with optimal transport costs and studies the connection between fair classification and regression.
result The optimal fair regression function is derived from the solution to a barycenter problem with optimal transport costs, and the optimal fair cost-sensitive classifiers can be derived by applying thresholds to this function.
Random scan CAVI converges linearly under log-concave assumptions.
problem Analyzing the convergence rate of random scan Coordinate Ascent Variational Inference (CAVI) under log-concave conditions.
method Building on previous work, we analyze the random scan version of CAVI using optimal transport geometry.
result We obtain tight linear convergence rates for the random scan version of CAVI.
Paper reformulates UOT as non-negative penalized linear regression for efficient algorithms.
problem Optimal transport with relaxed marginal conditions.
method Reformulate UOT as non-negative penalized linear regression, propose multiplicative updates.
result Efficient algorithms for UOT with quadratic penalties, continuity of solutions.
Sharp 2-Wasserstein bounds for DDPMs derived from Föllmer process.
problem Sampling error bounds for DDPMs in 2-Wasserstein distance.
method Lipschitz-type conditions on score function, Föllmer process, and log-concave target distributions.
result Sharp upper bounds for DDPMs in 2-Wasserstein distance, optimal in dimension and steps.