Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces

Nonlinear sufficient dimension reduction for distribution-on-distribution regression

We introduce a new approach to nonlinear sufficient dimension reduction in cases where both the predictor and the response are distributional data, modeled as members of a metric space. Our key step is to build universal kernels (cc-universal) on the metric spaces, which results in reproducing kernel Hilbert spaces for the predictor and response that are rich enough to characterize the conditional independence that determines sufficient dimension reduction. For univariate distributions, we construct the universal kernel using the Wasserstein distance, while for multivariate distributions, we resort to the sliced Wasserstein distance. The sliced Wasserstein distance ensures that the metric space possesses similar topological properties to the Wasserstein space, while also offering significant computation benefits. Numerical results based on synthetic data show that our method outperforms possible competing methods. The method is also applied to several data sets, including fertility and mortality data and Calgary temperature data.

On the capacity of deep generative networks for approximating distributions

We study the efficacy and efficiency of deep generative networks for approximating probability distributions. We prove that neural networks can transform a low-dimensional source distribution to a distribution that is arbitrarily close to a high-dimensional target distribution, when the closeness is measured by Wasserstein distances and maximum mean discrepancy. Upper bounds of the approximation error are obtained in terms of the width and depth of neural network. Furthermore, it is shown that the approximation error in Wasserstein distance grows at most linearly on the ambient dimension and that the approximation order only depends on the intrinsic dimension of the target distribution. On the contrary, when f-divergences are used as metrics of distributions, the approximation property is different. We show that in order to approximate the target distribution in f-divergences, the dimension of the source distribution cannot be smaller than the intrinsic dimension of the target distribution.

Entropy-Regularized Optimal Transport on Multivariate Normal and q-normal Distributions

The distance and divergence of the probability measures play a central role in statistics, machine learning, and many other related fields. The Wasserstein distance has received much attention in recent years because of its distinctions from other distances or divergences. Although computing the Wasserstein distance is costly, entropy-regularized optimal transport was proposed to computationally efficiently approximate the Wasserstein distance. The purpose of this study is to understand the theoretical aspect of entropy-regularized optimal transport. In this paper, we focus on entropy-regularized optimal transport on multivariate normal distributions and q-normal distributions. We obtain the explicit form of the entropy-regularized optimal transport cost on multivariate normal and q-normal distributions; this provides a perspective to understand the effect of entropy regularization, which was previously known only experimentally. Furthermore, we obtain the entropy-regularized Kantorovich estimator for the probability measure that satisfies certain conditions. We also demonstrate how the Wasserstein distance, optimal coupling, geometric structure, and statistical efficiency are affected by entropy regularization in some experiments. In particular, our results about the explicit form of the optimal coupling of the Tsallis entropy-regularized optimal transport on multivariate q-normal distributions and the entropy-regularized Kantorovich estimator are novel and will become the first step towards the understanding of a more general setting.