Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces
This work addresses theoretical convergence guarantees for empirical measures in high-dimensional and functional data analysis, which is incremental but important for statistical learning and probability theory applications.
The paper provides upper bounds for the expected Wasserstein distance between probability measures and their empirical versions in unbounded functional spaces, generalizing previous results from finite-dimensional Euclidean and bounded functional spaces with optimal dimensionality dependence. It achieves rate-optimal bounds for Gaussian processes in separable Hilbert spaces with geometrically or polynomially decaying coordinates, and combines these with concentration results to yield improved exponential tail probability bounds under Bernstein-type or log Sobolev-type conditions.
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.