Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces

Abstract

We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence on the dimensionality. Our method also covers the important case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper bounds for functional data distributions whose coordinates decay geometrically or polynomially. Moreover, our bounds of the expected value can be combined with mean-concentration results to yield improved exponential tail probability bounds for the Wasserstein error of empirical measures under Bernstein-type or log Sobolev-type conditions.

arXiv comment35 pages

References in corpus (8)

• Wasserstein GAN
• Properties of principal component methods for functional and longitudinal data analysis
• Rate-optimal graphon estimation
• Nonparametric graphon estimation
• Stochastic blockmodel approximation of a graphon: Theory and consistent estimation
• Concentration in unbounded metric spaces and algorithmic stability
• Minimax Distribution Estimation in Wasserstein Distance
• Network Representation Using Graph Root Distributions

Cited by in corpus (37)

• Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networks
• On the Complexity of Approximating Multimarginal Optimal Transport
• Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs
• Minimax Confidence Intervals for the Sliced Wasserstein Distance
• Sharp Convergence Rates for Empirical Optimal Transport with Smooth Costs
• On the capacity of deep generative networks for approximating distributions
• Evaluating Model Robustness and Stability to Dataset Shift
• Minimax Distribution Estimation in Wasserstein Distance
• Bootstrapping Networks with Latent Space Structure
• Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey
• Relaxed Wasserstein with Applications to GANs
• Network Representation Using Graph Root Distributions
• Statistical guarantees for generative models without domination
• Smooth $p$-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications
• Stein variational gradient descent on infinite-dimensional space and applications to statistical inverse problems
• A Riemannian Block Coordinate Descent Method for Computing the Projection Robust Wasserstein Distance
• Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applications
• Asymptotic Guarantees for Generative Modeling Based on the Smooth Wasserstein Distance
• Nonparametric two-sample hypothesis testing for low-rank random graphs of differing sizes
• Projection Robust Wasserstein Distance and Riemannian Optimization
• Unseeded low-rank graph matching by transform-based unsupervised point registration
• Two-sample Test with Kernel Projected Wasserstein Distance
• Convergence Rates for Empirical Measures of Markov Chains in Dual and Wasserstein Distances
• Randomised Wasserstein Barycenter Computation: Resampling with Statistical Guarantees
• On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification
• Continuity of Generalized Entropy and Statistical Learning
• Consistent polynomial-time unseeded graph matching for Lipschitz graphons
• A Nonparametric Bayesian Framework for Uncertainty Quantification in Stochastic Simulation
• Wasserstein Conditional Independence Testing
• Uniform Confidence Band for Optimal Transport Map on One-Dimensional Data
• Online Learning of Parameterized Uncertain Dynamical Environments with Finite-sample Guarantees
• Convergence of Gaussian-smoothed optimal transport distance with sub-gamma distributions and dependent samples
• A second order analysis of McKean-Vlasov semigroups
• A new approach to posterior contraction rates via Wasserstein dynamics
• Estimation and Quantization of Expected Persistence Diagrams
• High-Dimensional Non-Parametric Density Estimation in Mixed Smooth Sobolev Spaces
• Entropy-regularized optimal transport on multivariate normal and q-normal distributions