Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
Rigidity of Wasserstein spaces over Riemannian manifolds
problem Isometric rigidity of L2 Wasserstein spaces over Riemannian manifolds
method Showing L2 Wasserstein spaces are isometrically rigid if and only if their underlying manifolds do not admit a Euclidean de Rham factor
result Isometry of L2 Wasserstein spaces over non-Euclidean manifolds
Develops calculus on Wasserstein spaces for Riemannian manifolds.
problem Characterizing and understanding the geometry of Wasserstein spaces.
method Intrinsic formalism for topology, smooth structure, and Riemannian geometry of Wasserstein spaces.
result Wasserstein spaces of closed manifolds are geodesically convex.
The space of Gaussian measures on a Euclidean space is geodesically convex in the L2-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the L2-Wasserstein space, we manag…
New Sliced-Wasserstein distances for non-Euclidean data.
problem Computational burden of Wasserstein distance on non-Euclidean manifolds.
method Derive Sliced-Wasserstein distances and flows on Cartan-Hadamard manifolds.
result General constructions and non-parametric schemes for minimizing new distances.
WSFN overcomes saddle points for non-convex functionals in Wasserstein space.
problem Minimizing non-convex functionals over the Wasserstein space with saddle point avoidance.
method WSFN is a second-order method that preconditions the Wasserstein gradient to avoid saddle points.
result WSFN escapes saddle regions and reaches a global minimizer in polynomial time.
We study rays and co-rays in the Wasserstein space Pp(X) (p>1) whose ambient space X is a complete, separable, non-compact, locally compact length space. We show that rays in the Wasserstein space can be represented as probability measures concentrated on the set of rays in the ambient spac…
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…
Study of isometries in Wasserstein spaces under specific conditions.
problem Characterizing isometries of Wasserstein spaces.
method Analysis of optimal maps and properties of Riemannian manifolds.
result Invariant set of Dirac deltas under isometries and isometry groups coincide.
Researchers find Busemann functions in Wasserstein space, enabling efficient projections and distances.
problem Defining Busemann functions in Wasserstein space for efficient data projections and distances.
method Investigated existence and computation of Busemann functions in Wasserstein space, establishing closed-form expressions for specific cases.
result Explicit projection schemes for probability distributions on \(\mathbb{R}\) enable novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets.
The paper shows how heat flows and Wasserstein distances relate to space rigidity.
problem Understanding rigidity in Wasserstein contraction along heat flows.
method Establishing equivalence between rigidity and Bakry-Émery gradient estimates, applying results from Ambrosio-Brué-Semola and Han.
result Spaces with specific curvature bounds exhibit rigidity in Wasserstein contraction.
New method speeds up optimization over probability measures.
problem High computational overhead in optimizing probability measures.
method Randomized coordinate descent on Wasserstein space.
result Significant speedups over full-gradient methods.
The paper proves inequalities linking Wasserstein distances and eigenfunctions in RCD(K,∞) spaces.
problem Estimating Wasserstein distances and their bounds in RCD(K,∞) spaces.
method Similar techniques used to prove inequalities involving p-Wasserstein distances and Laplace eigenfunctions. result Proves a conjectured lower bound on p-Wasserstein distance between positive and negative parts of Laplace eigenfunctions. New hyperbolic sliced-Wasserstein distances derived for efficient comparison.
problem Efficient comparison of distributions in hyperbolic spaces.
method Projections on geodesics or horospheres to derive novel sliced-Wasserstein distances.
result Novel hyperbolic sliced-Wasserstein distances are more computationally efficient.
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.
Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.
problem Estimating the expected max-sliced Wasserstein distance between a probability measure and its empirical distribution.
method Banach space version and operator norm approach for upper bounds.
result Upper bounds for max-sliced Wasserstein distances are essentially matching and sharp up to a log factor.
We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.
Unified theory of optimal transport for random measures.
problem Statistical uncertainty in optimal transport.
method Constructing L2 over Wasserstein space for random probability measures. result Unified treatment of random optimal transport and principled inference.
New algorithm accelerates optimization on Riemannian manifolds, including Wasserstein space.
problem Accelerating optimization methods in Riemannian geometry.
method Dynamic stepsize algorithms on Riemannian manifolds with specific vector transport.
result First provable accelerated gradient method in Wasserstein space.
The paper extends graph-based semi-supervised learning to infinite-dimensional Wasserstein space.
problem Graph-based semi-supervised learning in high-dimensional data.
method Laplace Learning in the Wasserstein space, proving variational convergence and characterizing the Laplace-Beltrami operator.
result Consistent classification performance in high-dimensional settings.
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…
Study entropic regularization of Gaussian measures and processes on Hilbert space.
problem Regularizing 2-Wasserstein distance for infinite-dimensional Gaussian measures and processes.
method Minimum Mutual Information property, closed form formulas, Fréchet differentiability, Sinkhorn barycenter equation.
result Entropic 2-Wasserstein distance and Sinkhorn divergence are Fréchet differentiable in Hilbert space.
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.
Study shows k-NN classifier is not universally consistent on (0,1) but consistent on discrete and specific measure spaces.
problem Consistency of k-NN classifier under Wasserstein distance on measure spaces. method Analysis of k-NN classifier properties under Wasserstein distance, use of σ-finite metric dimension, geodesic structures of Wasserstein spaces. result Consistency of k-NN classifier on specific measure spaces (discrete, Gaussian, wavelet series) but not on (0,1). In this short note, we would like to give a construction of parallel transport for tangent cones lying in the interior of a geodesic in Wasserstein space. We give a complete proof for the linear part of the tangent space, and show that a construction for the full tangent cones follows from some natural lemmas on Wasser…
Upper bound for max-sliced 2-Wasserstein distance between measures.
problem Estimating distance between probability measures and their empirical counterparts.
method Same technique as previous work, upper bound approach.
result Upper bound for expected max-sliced 2-Wasserstein distance.
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.
Aggregates probability models using Wasserstein space and variational approach.
problem Model aggregation in the Wasserstein space of distributions.
method Data-driven calibration framework based on Γ-convergence. result Empirical minimizers converge to the minimizers of the actual problem.
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.
This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.
problem Analyzing modes of variation in datasets of probability measures.
method Geodesic Principal Component Analysis (GPCA) on Wasserstein space with neural networks.
result Identification of geodesic curves that capture modes of variation in probability distributions.
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.
LOT Wassmap speeds up Wasserstein space manifold learning.
problem Finding low-dimensional structures in Wasserstein space datasets.
method Linearized optimal transport and approximation schemes.
result LOT Wassmap provides accurate embeddings with computational efficiency.
WGANs use optimal 1-Wasserstein distance to generate distributions.
problem Characterize geometrical properties of generated distributions.
method Analyze WGANs in finite and asymptotic regimes, focusing on univariate latent space.
result WGANs can approach target distribution with optimal 1-Wasserstein distance as sample size increases.
We extend the geometric study of the Wasserstein space W(X) of a simply connected, negatively curved metric space X by investigating which pairs of boundary points can be linked by a geodesic, when X is a tree.
Vanilla GANs are connected to Wasserstein distance for better understanding.
problem Understanding the statistical properties of Vanilla GANs.
method Connecting Vanilla GANs to Wasserstein distance and proving an oracle inequality.
result An oracle inequality for Vanilla GANs in Wasserstein distance is obtained.
A new algorithm computes Wasserstein barycenters without entropic regularization.
problem Computing Wasserstein barycenters efficiently and accurately.
method Free-support algorithm based on particle flow and Riemannian geometry.
result The algorithm avoids entropic regularization and is computationally tractable.
Paper introduces new Gromov-type distances for comparing Gaussian mixture models.
problem Comparing distributions across different metric spaces using Gromov-Wasserstein distances.
method Incorporates invariance properties into MW2, introducing MGW2 and EW2.
result MGW2 and EW2 are efficient for estimating distances between GMMs in practical applications.
In the last couple of years, several adversarial attack methods based on different threat models have been proposed for the image classification problem. Most existing defenses consider additive threat models in which sample perturbations have bounded L_p norms. These defenses, however, can be vulnerable against advers…
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.
Neural Local Wasserstein Regression models distribution-on-distribution regression with flexible, localized transport maps.
problem Estimating distribution-on-distribution regression with global optimal transport maps or linearization limitations.
method Proposes Neural Local Wasserstein Regression, a flexible nonparametric framework using locally defined transport maps in Wasserstein space.
result Demonstrates effective capture of nonlinear and high-dimensional distributional relationships.
We study the Wasserstein natural gradient in parametric statistical models with continuous sample spaces. Our approach is to pull back the L2-Wasserstein metric tensor in the probability density space to a parameter space, equipping the latter with a positive definite metric tensor, under which it becomes a Riemanni…
Study convergence and approximations of entropic regularized Wasserstein distances for Gaussian and RKHS measures.
problem Convergence and approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
method Analysis of convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
result Strictly weaker convergence in 2-Sinkhorn divergence for Gaussian measures compared to exact 2-Wasserstein distance.
We propose a novel fused Gromov-Wasserstein alignment method to jointly learn the Hawkes processes in different event spaces, and align their event types. Given two Hawkes processes, we use fused Gromov-Wasserstein discrepancy to measure their dissimilarity, which considers both the Wasserstein discrepancy based on the…
This paper addresses Gaussian Process regression over probability measures, revealing a non-stationarity issue between Euclidean and Wasserstein kernels.
problem Non-stationarity issue between Euclidean and Wasserstein kernels in Gaussian Process regression over probability measures.
method Assuming Euclidean input space, applying algebraic transformation based on uncovered non-stationarity relationship to create a non-stationary and Wasserstein-based Gaussian Process model.
result An algebraic transformation simplifies learning a non-stationary Gaussian Process model over probability measures.
A method for fast estimation of Wasserstein distances using sliced Wasserstein distances.
problem Efficiently computing Wasserstein distances for multiple pairs of distributions.
method Regression on sliced Wasserstein distances to predict true Wasserstein distances.
result The proposed method provides a better approximation of Wasserstein distance than state-of-the-art models, especially in low-data regimes.
A new conformal prediction framework for graph-valued outputs using Z-Gromov-Wasserstein distances.
problem Lack of principled uncertainty quantification for graph-valued supervised prediction.
method Proposes a conformal prediction framework using Z-Gromov-Wasserstein distances for graph-valued outputs.
result Provides distribution-free coverage guarantees for graph-valued outputs.
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.
This paper bridges variational inference and Wasserstein gradient flows.
problem Combining variational inference and Wasserstein gradient flows for more efficient approximations.
method Recasting Bures-Wasserstein gradient flow as a Euclidean gradient flow and using path-derivative gradient estimator.
result A new gradient estimator for f-divergences that can be implemented using machine learning libraries.