New method trains normalizing flows using entropy-regularized transport.
problem Training continuous normalizing flows efficiently.
method Formulates flows as gradients of scalar potentials, training only these potentials.
result Trains normalizing flows without explicit flow computation during training.
This is the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimal transport in the space of probability measures characterizes lower bounds of the Ricci curvature. We then discuss geometric properties of general metric measure …
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.
Entropy regularized OT test assesses independence between samples.
problem Testing independence between two samples.
method Entropy regularized optimal transport.
result Non-asymptotic bounds for test statistic established.
Paper develops fast method for computing optimal transport.
problem Efficient computation of optimal transport distance between distributions.
method Entropy-regularized extragradient method for first-order optimization.
result Achieves state-of-the-art runtime guarantees and good numerical performance.
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.
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.
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.
Enhances flexibility in data reweighting with optimal transport and maximum entropy principles.
problem Adapting empirical distributions to predefined constraints on moments, tail behavior, etc.
method Nonparametric distributional constraints, maximum entropy principle, optimal transport.
result Maximum entropy weight adjusted empirical distribution close to a specified distribution in optimal transport metric.
Stability result for a popular algorithm in optimal transport.
problem Stability of the Iterative Proportional Fitting Procedure in time and metric.
method Uniform stability analysis in the 1-Wasserstein metric.
result Quantitative stability result for entropy-regularized Optimal Transport and Schrödinger bridges.
We investigate the use of entropy-regularized optimal transport (EOT) cost in developing generative models to learn implicit distributions. Two generative models are proposed. One uses EOT cost directly in an one-shot optimization problem and the other uses EOT cost iteratively in an adversarial game. The proposed gene…
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.
Differentiable PF via entropy-regularized OT for better inference.
problem Non-differentiability of traditional PF resampling methods.
method Entropy-regularized optimal transport for differentiable resampling.
result Convergent differentiable PF method with improved gradient estimates.
Framework for analyzing dynamic topological changes in point clouds using persistent homology and dynamic optimal transport.
problem Analyzing transient structural reorganizations during dynamic phase transitions in time-evolutionary point clouds.
method Hierarchical dynamic evaluation framework driven by topological and hypergraph reconstruction strategy.
result Combining transport-based alignment with multi-scale entropy diagnostics for dynamic topological analysis.
Paper proves generalized Talagrand inequality for Sinkhorn distance.
problem Proving a generalized Talagrand inequality for Sinkhorn distance.
method Using entropy power inequality and infinitesimal displacement convexity of optimal transport map.
result Extends previous results of Gaussian Talagrand inequality for Sinkhorn distance to strongly log-concave case.
We show that a certain entropy-like function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman's reduced volume.
New proof of energy functional monotonicity via geodesics in measure space.
problem Proving monotonicity of energy functional in generalized Ricci flow.
method Defining adapted cost functional, geodesics, and entropy functional.
result Monotonicity of cost along backwards heat flow and energy functional along generalized Ricci flow.
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.
New scalable algorithm for non-negative linear regression with entropy-regularized OT loss.
problem Generalizing task-specific linear models to broader applications.
method Sinkhorn-like scaling iterations for convex penalty and datafit terms.
result Simple multiplicative updates for various penalty and datafit terms.
Study on convergence rates for optimal transport with regularization.
problem Convergence analysis of divergence-regularized optimal transport.
method Novel methodology using quantization and martingale couplings.
result Sharp rates for various divergences and transport costs.
A practical algorithm improves approximate OT distances using quantization.
problem Substantial computational burden in computing OT distances for large samples.
method Introduces a quantization step to estimate OT distances between measures.
result The quantization step improves the performance of approximate solvers for entropy-regularized transport.
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 …
New bounds show empirical EOT adapts to simpler measure.
problem Statistical performance of empirical EOT estimators.
method Novel statistical bounds, empirical process theory, dual formulation.
result Empirical EOT and its unregularized version follow lower complexity adaptation.
New method for comparing different mass measures on tree structures using entropy partial transport.
problem Comparing nonnegative measures with different masses on tree structures.
method Entropy Partial Transport (EPT) on extended trees, regularized for fast computation and negative definiteness.
result First closed-form solution for unbalanced OT on tree structures.
Improved first-order algorithm for entropy regularized OT with faster convergence.
problem Solving entropy regularized optimal transport efficiently.
method Accelerated primal-dual stochastic mirror descent algorithm with variance reduction.
result Improved rate from O ~ ( n 2.5 / ε ) \widetilde{O}({n^{2.5}}/ε) O ( n 2.5 / ε ) to O ~ ( n 2 / ε ) \widetilde{O}({n^2}/ε) O ( n 2 / ε ) . 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.
Develops new synthetic Ricci flow concepts for metric measure spaces.
problem No specific problem stated; focuses on new mathematical concepts.
method Formulated in terms of dynamic convexity and local concavity of entropy, and global/short-time asymptotic transport cost estimates.
result Shows these properties characterise smooth (weighted) Ricci flows.
New method synthesizes and analyzes probability measures using entropy-regularized optimal transport.
problem Synthesize and analyze probability measures with entropy-regularized optimal transport.
method Entropy-regularized Wasserstein-2 cost and Sinkhorn divergence for synthesis and analysis.
result Computed barycentric coefficients and their stability for classification of corrupted point cloud data.
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…
New Langevin dynamics samples from entropy-regularized optimal transport.
problem Sampling from entropy-regularized optimal transport.
method Introduced analogous diffusion dynamics constrained to Π ( μ , ν ) Π(μ,ν) Π ( μ , ν ) . result Long-time limit is the unique solution of an entropic optimal transport problem.
Grogan et al [11,12] have recently proposed a solution to colour transfer by minimising the Euclidean distance L2 between two probability density functions capturing the colour distributions of two images (palette and target). It was shown to be very competitive to alternative solutions based on Optimal Transport for c…
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.
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.
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.
This work extends entropic optimal transport to non-product reference couplings, focusing on Gaussian cases.
problem Finding a diffuse coupling between two measures with non-product reference couplings.
method Reduction of the entropic optimal transport problem to a matrix optimization problem.
result Complete description of the solution for non-product reference couplings, including primal and dual variables.
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.
Framework uses optimal transport for neural architecture search.
problem Optimizing neural architectures in deep learning.
method Semi-discrete optimization using optimal transport.
result Gradient flow and minimizing movement scheme converge to reaction-diffusion equations.
A novel approach for semi-supervised learning using regularized optimal transport.
problem Improving model performance with unlabeled data.
method Regularized optimal transport between empirical measures for affinity matrix construction, incremental label propagation, and certainty score.
result Surpasses state-of-the-art results on 12 benchmark datasets.
Study optimal transport for stationary processes, estimating joinings and costs.
problem Optimal transport for stationary stochastic processes.
method Introduced estimators for optimal joinings and costs, established consistency and error rates.
result Consistent estimators of optimal joinings and costs under mild and stronger mixing assumptions.
The Sinkhorn flow is a gradient flow in a nonlocal Wasserstein geometry.
problem Entropy-regularized optimal transport and its applications.
method Thermodynamic interpretation of the Sinkhorn algorithm.
result The Sinkhorn flow is the gradient flow of entropy in a nonlocal Wasserstein geometry.
New algorithm enhances generative modeling for bounded domains.
problem Ad-hoc thresholding techniques for boundary enforcement in diffusion models.
method Reflected Schrödinger Bridge algorithm for entropy-regularized optimal transport.
result Generative modeling in diverse bounded domains with optimal transport properties.
Study compares synthetic and distributional Ricci curvature bounds.
problem Comparing synthetic and distributional approaches to lower Ricci curvature bounds.
method Analyzes synthetic via weak displacement convexity and distributional via non-negativity of Ricci-tensor.
result Distributional bounds imply entropy bounds for C 1 C^1 C 1 metrics and vice versa for C 1 , 1 C^{1,1} C 1 , 1 under convergence condition. Paper estimates EOT maps for non-compactly supported measures with subGaussian target.
problem Estimating EOT maps between non-compactly supported measures.
method Uses bias-variance decomposition, T1-transport inequalities, and concentration of measure results.
result Shows error decay rates for different cases of subGaussian measures.
Debiased Wasserstein barycenters improve on entropy regularization in OT.
problem Entropy regularization in OT introduces bias, leading to blurred barycenters.
method Propose debiased Wasserstein barycenters using Sinkhorn iterations.
result Debiased barycenters preserve fast Sinkhorn-like iterations without entropy smoothing bias.
New algorithm computes Schrödinger Bridge for unpaired data translation.
problem Computing optimal transport maps for unpaired data translation.
method Schrödinger Bridge Flow, a discretization of a flow of path measures.
result Eliminates the need to train multiple DDM-like models.
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.
A new approach for test-time adaptation detects and reacts to distribution shifts.
problem Improving test-time accuracy under distribution shifts.
method Online self-training with a detection tool based on entropy values and betting martingales.
result The classifier's entropy values match those of the source domain, building invariance to distribution shifts.
The objective in statistical Optimal Transport (OT) is to consistently estimate the optimal transport plan/map solely using samples from the given source and target marginal distributions. This work takes the novel approach of posing statistical OT as that of learning the transport plan's kernel mean embedding from sam…