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.
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.
Paper connects surface shape analysis and unbalanced optimal transport.
problem Computing the SRNF shape distance on piecewise linear surfaces.
method Characterizes SRNF shape distance as WFR distance pullback, proposes new algorithm for WFR distance computation.
result Direct computation of SRNF shape distance on piecewise linear surfaces.
Optimal algorithms for Riemannian optimization with reduced complexity.
problem Stochastic optimization on Riemannian manifolds with limited data.
method Zeroth-order Riemannian Averaging Stochastic Approximation algorithms using Riemannian moving-average estimators and novel geometric conditions.
result Achieves optimal sample complexities for generating approximate first-order stationary solutions.
This paper analyzes minibatch optimal transport distances and their applications.
problem Optimal transport distances are complex and impractical for large datasets.
method Extended analysis of minibatch optimal transport distances, focusing on various kernels and debiased functions.
result Minibatch optimal transport distances are unbiased estimators and have statistical and optimisation properties.
Neural framework for conditional OT maps learns from categorical and continuous variables.
problem Learning conditional optimal transport maps between complex distributions.
method Hypernetwork generates adaptive transport layer parameters based on conditioning variables.
result Our method outperforms simpler conditioning methods in comprehensive ablation studies.
Paper adapts numerical scheme for parallel transport on diffeomorphism manifolds.
problem Efficient computation of parallel transport on high-dimensional manifold-valued data.
method Adapts a numerical scheme for parallel transport to finite-dimensional manifolds of diffeomorphisms.
result Qualitative and quantitative analysis of scheme's behavior on high-dimensional manifolds.
We analyze errors in filtering algorithms using optimal transport.
problem Estimation errors in optimal transport-based filtering algorithms.
method Systematic analysis of estimation errors for conditional Brenier maps.
result Demonstrates effectiveness and practical potential of the optimal transport filtering algorithm.
Proposes a method to correct for covariate shift in meta-analysis of randomized trials.
problem Invalidation of standard IPD meta-analysis due to covariate shift across studies.
method Placebo-anchored transport framework that treats source-trial outcomes as proxy signals and target-trial placebo outcomes as gold labels.
result Yields target-identified effect estimates in connected targets and a principled screen--then--transport procedure in disconnected targets.
Deep learning models optimize urban transportation scheduling.
problem Optimizing transportation systems with complex dynamics and large data sets.
method Developed deep learning metamodels for simulators and reinforcement learning algorithms.
result Improved optimal scheduling of travelers on transportation networks.
A new method improves Bayesian filtering in nonlinear systems.
problem Bayesian filtering in nonlinear dynamical systems with non-Gaussian posteriors.
method Transport maps with block-triangular structure and gradient flows for MMD minimization.
result Accurate approximation of non-Gaussian posteriors without particle collapse.
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).
This paper develops optimal transport methods on the roto-translation group SE2.
problem Optimal transport on the roto-translation group SE2 for image analysis.
method Develops a computational framework for optimal transportation over Lie groups, focusing on SE2. Uses Sinkhorn-like algorithm with efficient distance approximations.
result Advances in image barycentric interpolation, orientation field interpolation, and Wasserstein flows on SE2.
Functional-analytic method for stochastic parallel transport in bundles.
problem Stochastic parallel transport in Hermitian bundles over Riemannian manifolds.
method Purely functional-analytic construction.
result Obtained a general Feynman-Kac formula in vector bundles.
Unified framework for ensemble transport-based smoothing of non-Gaussian time series.
problem Bayesian time series re-analysis with non-Gaussian distributions.
method Measure transport approach to derive consistent prior-to-posterior transformations.
result General ensemble framework for transport-based smoothing of state-space models.
New method estimates transport distances with low rank couplings.
problem Estimating optimal transport distances in high dimensions.
method Factored couplings with low transport rank regularization.
result Improves Wasserstein distances and optimal transport plans.
BM2 learns Schrödinger bridges using neural networks.
problem Learning dynamic transport maps between two distributions.
method Coupled Bridge Matching (BM2) with neural networks. result Preliminary theoretical analysis and numerical experiments show BM2's effectiveness. This study analyzes the transport map in deep neural networks to explain their performance.
problem Understanding the role of depth and interpretation of hidden layers in deep neural networks.
method Developed flow representation and transport analysis of DNNs, interpreting them as Euler broken lines.
result Deeper DAEs converge faster and extract better features, reducing entropy.
We develop a computationally efficient method to estimate Ollivier-Ricci curvature.
problem Computational infeasibility of evaluating Ollivier-Ricci curvature on large graphs.
method Derive explicit transfer moduli between OR and BF curvatures, construct lazy transport envelopes, and use cross-edge matching.
result Deterministic bounds for OR curvature parameterized by local graph combinatorics, reducing complexity to worst-case O(max_v deg(v)^1.5).
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.
New transforms improve signal classification and data analysis.
problem Improving signal classification and data analysis.
method Algebraic generative models and transport transforms.
result Classes of signals are transformed into convex sets, simplifying classification.
Optimal transport with scalar martingales defined over multiple periods.
problem Finding optimal transport plans with specific properties over multiple time periods.
method Introducing left-monotone transports and characterizing them through various properties.
result Left-monotone transports are unique under certain conditions and have specific order properties.
Sharp ABP estimate on metric spaces via optimal transport.
problem Sharp ABP estimate on metric measure spaces.
method Optimal transport theory.
result Established a sharp ABP estimate on metric measure spaces.
Geometric analysis on diffeomorphism groups for fluid dynamics and information geometry.
problem Geometric analysis of fluid flows and optimal mass transport.
method Review of metrics and topology on diffeomorphism groups.
result Introduction of new metrics and topology for diffeomorphism groups.
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.
Improved spatial distribution learning with Bayesian transport maps and parametric shrinkage.
problem Learning non-Gaussian spatial distributions with limited training data.
method Proposed ShrinkTM approach using Bayesian transport maps with parametric shrinkage.
result ShrinkTM outperforms existing BTM, especially with few training samples.
Researchers found counterexamples to conjectures about optimal transport maps on curved spaces.
problem Extending Caffarelli's contraction theorem to curved spaces.
method Constructing counterexamples to precise conjectures.
result Found counterexamples to Milman's conjectures about optimal transport maps on curved spaces.
Improved Sinkhorn algorithm for UOT with near-linear complexity.
problem Solving the entropic regularized Unbalanced Optimal Transport problem efficiently.
method Geometric convergence analysis of Sinkhorn updates and primal solution properties.
result Near-linear time complexity for finding ε-approximate UOT solutions. We introduce a new framework for optimal transport using Schatten-p regularization to recover low-rank structures.
problem Optimal transport problems with low-rank structure recovery.
method Schatten-p norm regularization to promote low-rank structure in transport maps and plans.
result Unified convex programs for low-rank structure recovery with theoretical guarantees and efficient algorithms.
Paper shows near-linear time approximation for optimal transport.
problem Computing optimal transport distances efficiently.
method New analysis of Sinkhorn iteration and greedy coordinate descent algorithm, Greenkhorn.
result Achieves near-linear time approximation for optimal transport distances.
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 …
This paper explores how entropic regularization improves Wasserstein estimators' performance.
problem Improving the approximation and estimation properties of Wasserstein estimators.
method Entropic regularization of optimal transport costs to smooth Wasserstein estimators.
result Entropic regularization can achieve comparable statistical performance to un-regularized estimators at lower computational cost.
A new metric for comparing probability measures on graphs, scalable and negative definite.
problem Optimal transport's high complexity and indefiniteness for kernel machines.
method Sobolev transport metric for graph metrics, closed-form formula, negative definiteness.
result Sobolev transport yields a scalable and negative definite metric.
Develops a new PAC-Bayesian bound for evaluating randomness in predictors.
problem Lack of a PAC-Bayesian bound for deterministic predictors and continuous loss functions.
method PAC-Bayesian transportation bound unifying PAC-Bayesian and chaining methods.
result First PAC-Bayesian bound relating risks of any two predictors by their distance.
Paper analyzes adversarial risk using optimal transport.
problem Poor performance of machine learning on adversarial data.
method Optimal transport perspective, optimal transport plans (couplings), convexity, smoothness assumptions.
result Fundamental limits on adversarial risk calculated for various datasets.
Introduces a new space of Radon measures for better understanding persistence diagrams.
problem Lack of optimal transport-based formalism for persistence diagrams.
method Formalizes persistence diagrams as Radon measures on the upper half plane via optimal partial transport.
result Characterizes convergence and barycenters of persistence diagrams.
A new CO-Optimal Transport method optimizes transport maps between samples and features.
problem Optimal transport's limitations when samples are on different spaces.
method COOT: a novel OT problem that optimizes transport maps between samples and features simultaneously.
result COOT leads to performance improvements over state-of-the-art methods in domain adaptation and co-clustering.
New framework formalizes estimating valid transport maps, revealing their statistical limits.
problem Estimating valid transport maps in generative modeling.
method Formalized a minimax framework for estimating valid transport maps.
result Estimating any valid transport map is as hard as estimating the optimal transport map under standard stability assumptions.
Paper introduces SGA for barycenter optimization in optimal transport.
problem Optimizing Wasserstein barycenter for discrete distributions.
method Sobolev gradient ascent algorithm tailored to Wasserstein geometry.
result SGA achieves convergence rate similar to subgradient descent.
Theoretical study on using optimal transport for domain adaptation.
problem Improving efficiency of machine learning algorithms in domain adaptation.
method Theoretical analysis of optimal transport theory applied to three DA settings.
result Wasserstein metric provides generalization guarantees for DA.
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.
The paper explores rectified flows and their relation to optimal transport.
problem Understanding the connection between rectified flows and optimal transport.
method Investigates invariance properties, explicit constructions, and analysis of rectified flows in various settings.
result Rectified flows, when gradient constrained, do not generally solve optimal transport problems.
Improves QMC for complex distributions using transport maps.
problem Challenges in applying QMC to general target distributions.
method Train a transport map to approximate target distributions, ensuring RQMC achieves superior error rates.
result Transport QMC achieves faster convergence rates than standard Monte Carlo under mild conditions.
The paper analyzes transportation dynamics of denoising autoencoders.
problem Analyzing the behavior of deep neural networks.
method Regarding feature maps as transport maps and analyzing transportation dynamics.
result The infinitely deep DAE transports mass to decrease entropy of data distribution.
Stochastic optimization improves semi-discrete OT map estimation with a minimax rate.
problem Empirical success of SGD in semi-discrete OT, but lack of theoretical guarantees.
method Averaged projected SGD with a minimax convergence rate of O(1/√n).
result SGD methods can estimate the OT map with a minimax convergence rate of O(1/√n).
Optimal transport with f-divergence regularization using generalized Sinkhorn algorithm.
problem Optimal transport with f-divergence regularization. method Generalized Sinkhorn algorithm for solving optimal transport problems with various f-divergences. result Strong duality holds, optimums are attained, and convergence to an optimal solution is guaranteed under certain conditions.
The paper analyzes rates of convergence for optimal transport map estimators using barycentric projections.
problem Estimating optimal transport maps from data sampled according to two distributions.
method Comprehensive analysis of rates of convergence for plug-in estimators defined via barycentric projections.
result New stability estimate for barycentric projections under minimal smoothness assumptions.
Proposes batch version of Greenkhorn for multimarginal OT, proving convergence.
problem Optimal transport problems with multiple marginals.
method Batch Greenkhorn algorithm, iterative Bregman projections, greedy control.
result Global linear rate of convergence and explicit iteration complexity bounds.