New algorithm approximates continuous Wasserstein barycenters efficiently.
problem Computing Wasserstein barycenters for continuous distributions.
method Stochastic algorithm using dual potentials and stochastic gradient descent.
result Efficient online approximation of continuous Wasserstein barycenters.
Develops a method to efficiently compute Wasserstein barycenters with variational distributions.
problem High computational burden in computing Wasserstein barycenters for high-dimensional and continuous settings.
method Introduces a variational distribution to approximate the continuous Wasserstein barycenter, reformulating the problem as an optimization with c-cyclical monotonicity.
result The method provides a tractable dual formulation for efficient computation of Wasserstein barycenters, demonstrated on real applications.
A neural network speeds up computation of Wasserstein barycenters by 60x.
problem Computing Wasserstein barycenters is computationally demanding.
method Trained a deep convolutional neural network to compute Wasserstein barycenters.
result Computational times reduced from milliseconds to seconds.
Estimates barycenter in geodesic spaces with finite sample bounds.
problem Estimating the barycenter of a distribution in geodesic spaces.
method Finite sample error bounds, Hoeffding- and Bernstein-type concentration inequalities, efficient algorithms.
result Statistical guarantees for efficient barycenter computation.
Paper solves barycenter of probability distributions using Sinkhorn divergence.
problem Computing the barycenter of a set of probability distributions under the Sinkhorn divergence.
method Recast as unconstrained functional optimization and develop Sinkhorn Descent (SD) method.
result SD converges to a stationary point at a sublinear rate and asymptotically finds a global minimizer.
Proposes variational Wasserstein barycenters for geometric clustering.
problem Geometric clustering problems, especially K-means and co-clustering.
method Solves for Monge maps using variational principle, explores connections to K-means and co-clustering.
result Demonstrates feasibility and use of variational Wasserstein barycenters in clustering.
New method for robustly estimating barycenters in data aggregation.
problem Outliers and noise in data measures hinder traditional OT barycenter estimation.
method Proposes a novel scalable approach using semi-unbalanced neural optimal transport.
result Demonstrates robustness to outliers and class imbalance.
Paper proposes PRWB and RPRWB for Wasserstein barycenters.
problem Numerical challenges in computing Wasserstein barycenters.
method Projection robust Wasserstein barycenter (PRWB) and relaxed PRWB (RPRWB).
result RPRWB improves clustering performance on real text datasets.
We present a stochastic algorithm to compute the barycenter of a set of probability distributions under the Wasserstein metric from optimal transport. Unlike previous approaches, our method extends to continuous input distributions and allows the support of the barycenter to be adjusted in each iteration. We tackle the…
Proves uniqueness of barycenters on manifolds without restrictions.
problem Finding unique barycenters on complex geometric spaces.
method Introduces new disintegrated Monge-Kantorovich metrics for barycenter problems.
result Uniqueness of barycenters on connected, complete Riemannian manifolds.
Efficiently computes optimal transport maps and Wasserstein barycenters using conditional normalizing flows.
problem Computing optimal transport maps and Wasserstein barycenters in high-dimensional spaces.
method Uses conditional normalizing flows to approximate distributions and solve the primal problem.
result Shows computational feasibility for hundreds of input distributions and yields accurate results.
Efficient federated algorithm for calculating transportation barycenter.
problem Efficiently calculating the free-support transportation barycenter in a federated setting.
method Single-loop dual decomposition algorithm that uses only aggregated information.
result Significantly scalable and low-complexity algorithm for federated computation.
New algorithm computes optimal transport barycenter efficiently.
problem Computing optimal transport barycenter for high-dimensional probability distributions.
method Wasserstein-Descent H˙1-Ascent (WDHA) algorithm. result Exact barycenter computation in nearly linear time and linear space complexity.
New scalable algorithm estimates barycenters of measures in high dimensions.
problem Estimating barycenters of measures in high-dimensional settings.
method Optimizes generative models to estimate barycenters, scaling by introducing inductive biases.
result First scalable method to estimate barycenters in thousands of dimensions.
We study in this paper a variant of Wasserstein barycenter problem, which we refer to as tree-Wasserstein barycenter, by leveraging a specific class of ground metrics, namely tree metrics, for Wasserstein distance. Drawing on the tree structure, we propose an efficient algorithmic approach to solve the tree-Wasserstein…
Proposes MFSWB for marginal fairness in SWB, improving efficiency and performance.
problem Achieving marginal fairness in SWB averaging.
method Defining MFSWB as a constrained SWB problem, proposing two surrogate problems and a new slicing distribution.
result Surrogate MFSWB problems effectively minimize distances to marginals and encourage marginal fairness.
We consider the problem of computing a Wasserstein barycenter for a set of discrete probability distributions with finite supports, which finds many applications in areas such as statistics, machine learning and image processing. When the support points of the barycenter are pre-specified, this problem can be modeled a…
A new algorithm for estimating continuous entropic barycenters under arbitrary costs.
problem Estimating the average of probability distributions under arbitrary cost functions.
method Dual reformulation of Entropic Optimal Transport (EOT) problem based on weak OT.
result Established quality bounds for the recovered solution and seamless integration with EBM learning.
New barycenters defined for hyperbolic balls, differing from spheres.
problem Defining barycenters in hyperbolic geometry.
method Introducing conformal and holomorphic barycenters.
result Holomorphic and conformal barycenters differ in hyperbolic balls.
New method for scalable barycenter computation using Wasserstein gradient flows.
problem Scalability and integration of label information in barycenter computation.
method Gradient flows in Wasserstein space, time discretization, mini-batch optimal transport, modular regularization, task-aware functions, supervised information integration.
result Empirically validated new state-of-the-art barycenter solver with labeled barycenters outperforming unlabeled ones.
We solve a complex optimization problem for Wasserstein barycenters using stochastic methods.
problem Optimizing the average of multiple probability distributions in a streaming data setting.
method We reformulate the problem as a convex-concave saddle-point problem and propose a stochastic optimization algorithm.
result Our algorithm has better complexity than existing methods for arbitrary distributions.
Paper tackles robust optimal transport with improved computational complexity and barycenter approximation.
problem Computing robust optimal transport and its barycenter efficiently.
method Sinkhorn-based algorithms for robust optimal transport and iterative Bregman projections for barycenter approximation.
result Improved computational complexity for robust optimal transport and barycenter approximation.
In the machine learning and optimization community, there are two main approaches for the convex risk minimization problem, namely, the Stochastic Approximation (SA) and the Sample Average Approximation (SAA). In terms of oracle complexity (required number of stochastic gradient evaluations), both approaches are consid…
Efficiently aggregating data from different sources is a challenging problem, particularly when samples from each source are distributed differently. These differences can be inherent to the inference task or present for other reasons: sensors in a sensor network may be placed far apart, affecting their individual meas…
New algorithm for computing Wasserstein barycenters with guarantees.
problem Computing Wasserstein barycenters with varying regularization strengths.
method Damped Sinkhorn iterations followed by exact maximization/minimization steps.
result First non-asymptotic convergence guarantees for approximating Wasserstein barycenters.
This work tackles regression on non-Euclidean spaces, specifically positive-definite matrices with the Bures-Wasserstein metric.
problem Regression on non-Euclidean spaces, specifically positive-definite matrices with the Bures-Wasserstein metric.
method Developed a sufficient condition for the existence of a minimizer of the conditional barycenter problem, characterized the optimization landscape, and developed a projection-free algorithm for approximate computation of first-order stationary points.
result The objective is free of local maxima under the sufficient condition, and the algorithm enables the use of stochastic Riemannian optimization methods for large-scale setups.
Algorithm samples from Wasserstein barycenter of measures.
problem Sampling from Wasserstein barycenter of measures.
method Gradient flow of multimarginal formulation with penalization.
result Algorithm samples close to Wasserstein barycenter.
We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We propose two original algorithms to compute Wasse…
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.
The paper extends a prediction method to curved spaces.
problem Prediction with expert advice in curved spaces.
method Geometric considerations and adaptation of online to batch conversion.
result Extension of exponentially weighted average forecaster to non-positive curvature spaces.
The paper develops a method to compute the conformal barycenter in hyperbolic space.
problem Computing the conformal barycenter in hyperbolic space.
method Analysis of Riemannian Newton's method and regularized Newton's method with line search.
result Newton's method and regularized Newton's method with line search converge to the conformal barycenter.
New reformulations for multiclass classification problems using optimal transport.
problem Adversarial multiclass classification problems.
method Multimarginal optimal transport formulation.
result Reveals geometric structure and extends binary classification results.
First DP algorithm for Wasserstein barycenters on private data.
problem Computing Wasserstein barycenters on private datasets.
method Differentially private algorithms for Wasserstein barycenters.
result High-quality private barycenters with strong accuracy-privacy tradeoffs.
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.
A new method for barycenter of probability measures using entropic optimal transport.
problem Finding a weighted average of probability distributions.
method Doubly regularized Wasserstein barycenters with entropic optimal transport.
result The new formulation is debiased and has a smooth density, leading to efficient estimation and optimization.
We consider a free boundary problem for the Willmore functional. Given a smooth domain Ω in R3, we construct Willmore disks wich are critical in the class of surfaces meeting ∂Ω orthogonally along their boundary and having small prescribed area. Using rescaling we first obtain constrained solut…
Paper introduces a novel framework for supervised graph prediction using Optimal Transport.
problem Supervised labeled graph prediction.
method Fused Gromov-Wasserstein (FGW) loss and FGW barycenter with neural network weights and learned graphs.
result The method can interpolate in the labeled graph space and achieve good performance on difficult problems.
Method approximates Riemannian barycenter on manifolds.
problem Computing the exact Riemannian barycenter is computationally expensive.
method Uses under- and over-approximations of Riemannian distance to compute an approximate barycenter.
result Approximation method is more efficient than exact methods and steepest descent.
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.
Study absolute continuity of Wasserstein barycenters on manifolds with singular cost functions.
problem Absolute continuity of Wasserstein barycenters on manifolds with singular cost functions.
method Approximation framework to handle singularity, geometrically transparent.
result Precise analytic condition on cost profile for necessary assumptions.
Study on Wasserstein barycenters with computational hardness and fast algorithm development.
problem Computing Wasserstein barycenters of discrete probability measures with fixed support.
method Developed a deterministic variant of IBP algorithm, FastIBP, with improved complexity.
result Demonstrated favorable performance of FastIBP in practice.
Paper presents a new algorithm to approximate Wasserstein-2 barycenters without bias.
problem Approximating Wasserstein-2 barycenters of continuous measures.
method Generative model approach using arbitrary neural networks.
result The method does not introduce bias and is applicable to large-scale tasks.
New dynamical approach defines symmedian as hyperbolic barycenter.
problem Understanding symmedian properties in hyperbolic geometry.
method Developed a new dynamical coordinatization.
result Symmedian point acts as hyperbolic barycenter.
A new method for averaging probability distributions based on optimal weak mass transport.
problem Averaging probability distributions in a geometric way.
method Weak barycenters based on optimal weak mass transport.
result Extracts common geometric information shared by all input distributions.
Proposes using Wasserstein barycenter for better multilingual alignment.
problem Finding word-to-word translations between multiple languages without parallel data.
method Uses Wasserstein barycenter as a more informative pivot language, minimizing pairwise transportation costs.
result Demonstrates state-of-the-art performances on standard benchmarks.
ScoreFusion fuses multiple diffusion models to enhance generative modeling of a target population.
problem Enhancing generative modeling of a target population with limited data.
method ScoreFusion uses KL barycenters of auxiliary populations and recasts the learning problem as score matching in denoising diffusion.
result ScoreFusion achieves a dimension-free sample complexity bound in total variation distance.
New method proves absolute continuity of Wasserstein barycenters on manifolds with lower Ricci curvature bound.
problem Proving absolute continuity of Wasserstein barycenters on manifolds with lower Ricci curvature bound.
method Introducing new displacement functionals exploiting Hessian equality and revisiting Souslin space theory, Dunford-Pettis theorem, and de la Vallée Poussin criterion for uniform integrability.
result Absolute continuity of Wasserstein barycenters is established for a general class of manifolds with lower Ricci curvature bound.
A scalable algorithm approximates Wasserstein Barycenters using neural networks.
problem Representing the weighted mean of probability distributions in high dimensions.
method Input Convex Neural Networks (ICNNs) for Kantorovich dual formulation of Wasserstein-2 distance.
result Generative model representation of the Barycenter with infinite samples.