The paper studies scaling limits of Wasserstein metrics on Gaussian mixture models.
problem Understanding the scaling limits of Wasserstein metrics on Gaussian mixture models.
method Scaling limit approach on Gaussian mixture models, including inhomogeneous and extended models.
result Existence of the limit of the Wasserstein metric after renormalization for GMMs with zero variance.
New Fourier metrics equivalent to Wasserstein distances in image processing.
problem Equivalence of Fourier-based and Wasserstein metrics in imaging problems.
method Extensions of Fourier-based metrics to handle different centers of mass and discrete measures, showing equivalence to Wasserstein distances.
result New Fourier metrics are equivalent to Wasserstein distances with explicit constants, improving runtime in image processing.
Permutation invariant network learns Wasserstein metrics.
problem Understanding the space of probability measures and comparing distributions.
method Permutation invariant network mapping samples to a low-dimensional space.
result Network can generalize to compute distances between unseen densities and learn moments.
We study the minimax optimal rate for estimating the Wasserstein-1 metric between two unknown probability measures based on n i.i.d. empirical samples from them. We show that estimating the Wasserstein metric itself between probability measures, is not significantly easier than estimating the probability measures u…
Introduces new metric for Riemannian metrics, extending unbalanced optimal transport.
problem Extending unbalanced optimal transport to Riemannian metrics.
method Dynamic and static formulations of unbalanced optimal transport on Riemannian metrics.
result Wasserstein--Ebin metric provides a new Riemannian structure on the space of Riemannian metrics.
We present a framework for Nesterov's accelerated gradient flows in probability space to design efficient mean-field Markov chain Monte Carlo (MCMC) algorithms for Bayesian inverse problems. Here four examples of information metrics are considered, including Fisher-Rao metric, Wasserstein-2 metric, Kalman-Wasserstein m…
A new supervised tree-Wasserstein distance improves document classification.
problem Measuring document similarity efficiently and accurately.
method Rewriting Wasserstein distance on tree metric, using contrastive loss for optimization.
result The Supervised Tree-Wasserstein (STW) distance improves document classification accuracy.
Generative Adversial Networks (GANs) have made a major impact in computer vision and machine learning as generative models. Wasserstein GANs (WGANs) brought Optimal Transport (OT) theory into GANs, by minimizing the 1-Wasserstein distance between model and data distributions as their objective function. Since then, W…
Study on conditions for achieving optimal robustness in statistical estimators.
problem Achieving the optimal robustness of estimators in statistical models.
method Developed a Wasserstein analogue of the Cramer-Rao inequality and investigated conditions for achieving the Wasserstein-Cramer-Rao lower bound.
result Conditions for the existence of asymptotically efficient estimators in one-parameter models and location-scale families.
Our attacks are stronger and faster under Wasserstein metric.
problem Vulnerability of deep models to adversarial attacks.
method Developed an exact yet efficient projection operator and used the Frank-Wolfe method.
result Generated much stronger attacks and improved model robustness.
Study geodesic properties of time series data using Wasserstein metric.
problem Modeling nonlinear time series with transport-based metrics.
method Generalized Wasserstein metric and signed cumulative distribution transforms.
result Geodesic properties provide added interpretability and robustness in time series classifiers.
A new robust metric compares distributions more accurately than existing methods.
problem Sensitivity to outliers and sampling discrepancy in Wasserstein distances.
method Introducing k-RPW, a partial p-Wasserstein distance.
result k-RPW converges faster to true distance and is more robust to outliers.
This paper develops efficient bounds on the Wasserstein metric for discrete measures.
problem Computing the exact Wasserstein metric is computationally expensive.
method Formulates and solves a Kantorovich problem on a coarse grid using quantized measures and cost matrices, followed by upscaling and correction.
result Achieves a 10x-100x speedup while maintaining low approximation error.
The Wasserstein probability metric has received much attention from the machine learning community. Unlike the Kullback-Leibler divergence, which strictly measures change in probability, the Wasserstein metric reflects the underlying geometry between outcomes. The value of being sensitive to this geometry has been demo…
This paper studies geometric properties of Wasserstein metric on SPD(n).
problem Understanding the geometry of symmetric positive-definite matrices under Wasserstein metric.
method Using fiber bundles, the paper derives explicit geometric quantities and proves global properties.
result The manifold is globally geodesic convex with non-negative curvatures but no conjugate pair and cut locus.
New algorithms solve partial optimal transport problems for applications like PU learning.
problem Optimal transport constraints on equal mass distributions limit applicability.
method Developed exact algorithms for partial Wasserstein and Gromov-Wasserstein problems.
result Partial Wasserstein metrics show effectiveness in positive-unlabeled learning.
A new portfolio model improves on Kelly's by accounting for estimation error.
problem Estimation error in Kelly portfolio optimization.
method Wasserstein distributionally robust optimization (DRO) to define a robust log-optimal portfolio.
result The Wasserstein-Kelly portfolio outperforms the Kelly portfolio in out-of-sample testing.
New Gromov-Wasserstein metric controls rigidity and incorporates prior knowledge.
problem Inflexible Gromov-Wasserstein distance and lack of feature alignment.
method Augmented Gromov-Wasserstein distance with feature alignments and prior knowledge.
result Improved performance in single-cell multi-omic alignment and transfer learning.
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…
Improved GAN performance using higher-order Wasserstein moments.
problem Stabilizing and enhancing GANs for better mode coverage and stability.
method Deriving and training a GAN with a modified Wasserstein distance using higher-order moments.
result Training a GAN with higher-order Wasserstein moments improves performance, even with increased computational cost.
A new method for Gaussian filtering using gradient flows and Wasserstein metrics.
problem Approximating Gaussian and mixture-of-Gaussians filtering for complex systems.
method Variational approximation via gradient-flow representation on Wasserstein metric space.
result Competitive performance in posterior representation and parameter estimation for systems with multiplicative noise and multi-modal distributions.
Learning to predict multi-label outputs is challenging, but in many problems there is a natural metric on the outputs that can be used to improve predictions. In this paper we develop a loss function for multi-label learning, based on the Wasserstein distance. The Wasserstein distance provides a natural notion of dissi…
Modified Wasserstein metric for Gaussian distributions, invariant to isometries.
problem Distance measurement for latent Gaussian distributions invariant to isometries.
method Modified Benamou-Brenier approach leading to a Procrustes Wasserstein metric.
result For Gaussian distributions, the metric reduces to Euclidean distance between eigenvalues.
Euclidean embeddings of data are fundamentally limited in their ability to capture latent semantic structures, which need not conform to Euclidean spatial assumptions. Here we consider an alternative, which embeds data as discrete probability distributions in a Wasserstein space, endowed with an optimal transport metri…
The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on statistical minimax rates for the problem of estimating a probability distribution under W…
Proposes a new distance metric for multi-marginal optimal transport.
problem Computational scalability in multi-marginal optimal transport.
method Random one-dimensional projections to construct sliced multi-marginal Wasserstein distance.
result Sliced multi-marginal Wasserstein distance is a metric with dimension-free sample complexity.
A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first part, we generalize the Hausdorff dimension by defining a family of bi-Lipschitz inv…
Develops a two-sample test using projected Wasserstein distance to handle high-dimensional data.
problem Testing whether two high-dimensional samples come from the same distribution.
method Optimal projection to find a low-dimensional linear mapping that maximizes the Wasserstein distance between projected probability distributions.
result Characterizes the convergence rate of the projected Wasserstein distance and presents practical algorithms.
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.
New statistical methods for analyzing distributions using Wasserstein metric.
problem Statistical analysis of probability distributions on the real line.
method Projected methods exploiting Wasserstein metric and Riemannian structure.
result Projected PCA and regression methods are faster and more flexible.
New slicing methods speed up Gaussian mixture Wasserstein distance computations.
problem High computational cost of the mixture Wasserstein distance.
method Slicing-based approximations to reduce computational complexity.
result Significant reduction in computational complexity while preserving key properties.
New method for reducing dimensions of distributional data.
problem Nonlinear sufficient dimension reduction for distribution-on-distribution regression.
method Building universal kernels on metric spaces to characterize conditional independence.
result Method outperforms competing methods in synthetic and real data applications.
New tools for estimating and inferring Wasserstein distance in topic models.
problem Estimating and inferring the Wasserstein distance between mixing measures in topic models.
method New canonical interpretation and tools for inference on Wasserstein distance in topic models.
result First minimax lower bounds and fully data-driven inferential tools for the Wasserstein distance in topic models.
We introduce a framework for Newton's flows in probability space with information metrics, named information Newton's flows. Here two information metrics are considered, including both the Fisher-Rao metric and the Wasserstein-2 metric. A known fact is that overdamped Langevin dynamics correspond to Wasserstein gradien…
Improved robustness in multivariate regression and classification with DRO under Wasserstein metric.
problem Outliers in covariates and responses.
method Distributionally Robust Optimization (DRO) with Wasserstein metric ambiguity set and regularization.
result Significant improvement in predictive error and robustness.
The paper examines rigidity of metric constructions in Wasserstein spaces.
problem Isometric rigidity of metric constructions in Wasserstein spaces.
method Analyzes spaces like Hilbert, rays, half-cylinders, and spherical suspensions.
result Different spaces exhibit varying levels of isometric rigidity in Wasserstein spaces.
Paper optimizes classification of distributions using Wasserstein metric.
problem Classifying instances represented by distributions on a vector space.
method Maximizing Fisher's ratio in the Wasserstein metric space through iterative algorithm.
result The method enhances classification performance and is robust to variations in distribution summaries.
We propose a framework, named Aggregated Wasserstein, for computing a dissimilarity measure or distance between two Hidden Markov Models with state conditional distributions being Gaussian. For such HMMs, the marginal distribution at any time spot follows a Gaussian mixture distribution, a fact exploited to softly matc…
New method improves Wasserstein distance for large-scale data.
problem High computational cost of Wasserstein distance for large-scale machine learning.
method Augmented Sliced Wasserstein Distances (ASWDs) using neural network mappings.
result ASWDs significantly outperform other Wasserstein variants in synthetic and real-world problems.
GWCA analyzes cross-graph correlations for movie retrieval.
problem Cross heterogeneous graph comparison in movie retrieval.
method Spectral graph filtering, Wasserstein metric learning, generalized eigenvalue decomposition.
result Surprise consistency in learning processes and closed-form solution.
Introduces new Wasserstein distances for more intrinsic metrics.
problem Improve metric for comparing distributions.
method Introduces RWp distances, designs algorithms for computation. result New distances are more intrinsic and computable.
Improves point-cloud reconstruction by optimizing projections with self-attention.
problem Inefficient and non-metric projection methods for sliced Wasserstein distances.
method Proposes distributional sliced Wasserstein distance with self-attention for permutation-invariant and metric optimization.
result Self-attention amortized distributional projection optimization achieves better performance in point-cloud reconstruction.
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.
Unsupervised learning of disentangled representations involves uncovering of different factors of variations that contribute to the data generation process. Total correlation penalization has been a key component in recent methods towards disentanglement. However, Kullback-Leibler (KL) divergence-based total correlatio…
Implementing k-NN classification using Gromov--Wasserstein distances
problem Comparing metric measure spaces
method Gromov--Wasserstein and fused Gromov--Wasserstein distances
result Universal consistency of k-NN classifiers A new Wasserstein K-means method for clustering probability distributions.
problem Clustering probability distributions using the Wasserstein metric.
method Distance-based K-means with SDP relaxation for Wasserstein barycenters. result Distance-based K-means outperforms centroid-based K-means for clustering probability distributions. We propose Gaussian optimal transport for Image style transfer in an Encoder/Decoder framework. Optimal transport for Gaussian measures has closed forms Monge mappings from source to target distributions. Moreover interpolates between a content and a style image can be seen as geodesics in the Wasserstein Geometry. Usi…
We propose a framework, named Aggregated Wasserstein, for computing a dissimilarity measure or distance between two Hidden Markov Models with state conditional distributions being Gaussian. For such HMMs, the marginal distribution at any time position follows a Gaussian mixture distribution, a fact exploited to softly …