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We analyze statistical properties of plug-in estimators for unbalanced optimal transport quantities between finitely supported measures in different prototypical sampling models. Specifically, our main results provide non-asymptotic bounds on the expected error of empirical Kantorovich-Rubinstein (KR) distance, plans, and barycenters for mass penalty parameter $C>0$. The impact of the mass penalty parameter $C$ is studied in detail. Based on this analysis, we mathematically justify randomized computational schemes for KR quantities which can be used for fast approximate computations in combination with any exact solver. Using synthetic and real datasets, we empirically analyze the behavior of the expected errors in simulation studies and illustrate the validity of our theoretical bounds., Comment: 61 pages

We provide a unifying approach to central limit type theorems for empirical optimal transport (OT). In general, the limit distributions are characterized as suprema of Gaussian processes. We explicitly characterize when the limit distribution is centered normal or degenerates to a Dirac measure. Moreover, in contrast to recent contributions on distributional limit laws for empirical OT on Euclidean spaces which require centering around its expectation, the distributional limits obtained here are centered around the population quantity, which is well-suited for statistical applications. At the heart of our theory is Kantorovich duality representing OT as a supremum over a function class $\mathcal{F}_{c}$ for an underlying sufficiently regular cost function $c$. In this regard, OT is considered as a functional defined on $\ell^{\infty}(\mathcal{F}_{c})$ the Banach space of bounded functionals from $\mathcal{F}_{c}$ to $\mathbb{R}$ and equipped with uniform norm. We prove the OT functional to be Hadamard directional differentiable and conclude distributional convergence via a functional delta method that necessitates weak convergence of an underlying empirical process in $\ell^{\infty}(\mathcal{F}_{c})$. The latter can be dealt with empirical process theory and requires $\mathcal{F}_{c}$ to be a Donsker class. We give sufficient conditions depending on the dimension of the ground space, the underlying cost function and the probability measures under consideration to guarantee the Donsker property. Overall, our approach reveals a noteworthy trade-off inherent in central limit theorems for empirical OT: Kantorovich duality requires $\mathcal{F}_{c}$ to be sufficiently rich, while the empirical processes only converges weakly if $\mathcal{F}_{c}$ is not too complex., Comment: 29 pages, 1 figure

The purpose of this paper is to provide a systematic discussion of a generalized barycenter based on a variant of unbalanced optimal transport (UOT) that defines a distance between general non-negative, finitely supported measures by allowing for mass creation and destruction modeled by some cost parameter. They are denoted as Kantorovich-Rubinstein (KR) barycenter and distance. In particular, we detail the influence of the cost parameter to structural properties of the KR barycenter and the KR distance. For the latter we highlight a closed form solution on ultra-metric trees. The support of such KR barycenters of finitely supported measures turns out to be finite in general and its structure to be explicitly specified by the support of the input measures. Additionally, we prove the existence of sparse KR barycenters and discuss potential computational approaches. The performance of the KR barycenter is compared to the OT barycenter on a multitude of synthetic datasets. We also consider barycenters based on the recently introduced Gaussian Hellinger-Kantorovich and Wasserstein-Fisher-Rao distances.

For probability measures on countable spaces we derive distributional limits for empirical entropic optimal transport quantities. More precisely, we show that the empirical optimal transport plan weakly converges to a centered Gaussian process and that the empirical entropic optimal transport value is asymptotically normal. The results are valid for a large class of cost functions and generalize distributional limits for empirical entropic optimal transport quantities on finite spaces. Our proofs are based on a sensitivity analysis with respect to norms induced by suitable function classes, which arise from novel quantitative bounds for primal and dual optimizers, that are related to the exponential penalty term in the dual formulation. The distributional limits then follow from the functional delta method together with weak convergence of the empirical process in that respective norm, for which we provide sharp conditions on the underlying measures. As a byproduct of our proof technique, consistency of the bootstrap for statistical applications is shown., Comment: 68 pages

Empirical optimal transport (OT) plans and distances provide effective tools to compare and statistically match probability measures defined on a given ground space. Fundamental to this are distributional limit laws and we derive a central limit theorem for the empirical OT distance of circular data. Our limit results require only mild assumptions in general and include prominent examples such as the von Mises or wrapped Cauchy family. Most notably, no assumptions are required when data are sampled from the probability measure to be compared with, which is in strict contrast to the real line. A bootstrap principle follows immediately as our proof relies on Hadamard differentiability of the OT functional. This paves the way for a variety of statistical inference tasks and is exemplified for asymptotic OT based goodness of fit testing for circular distributions. We discuss numerical implementation, consistency and investigate its statistical power. For testing uniformity, it turns out that this approach performs particularly well for unimodal alternatives and is almost as powerful as Rayleigh's test, the most powerful invariant test for von Mises alternatives. For regimes with many modes the circular OT test is less powerful which is explained by the shape of the corresponding transport plan., Comment: 24 pages, 9 figures

We consider a general linear program in standard form whose right-hand side constraint vector is subject to random perturbations. This defines a stochastic linear program for which, under general conditions, we characterize the fluctuations of the corresponding empirical optimal solution by a central limit-type theorem. Our approach relies on the combinatorial nature and the concept of degeneracy inherent in linear programming, in strong contrast to well-known results for smooth stochastic optimization programs. In particular, if the corresponding dual linear program is degenerate the asymptotic limit law might not be unique and is determined from the way the empirical optimal solution is chosen. Furthermore, we establish consistency and convergence rates of the Hausdorff distance between the empirical and the true optimality sets. As a consequence, we deduce a limit law for the empirical optimal value characterized by the set of all dual optimal solutions which turns out to be a simple consequence of our general proof techniques. Our analysis is motivated from recent findings in statistical optimal transport that will be of special focus here. In addition to the asymptotic limit laws for optimal transport solutions, we obtain results linking degeneracy of the dual transport problem to geometric properties of the underlying ground space, and prove almost sure uniqueness statements that may be of independent interest.

We derive limit distributions for certain empirical regularized optimal transport distances between probability distributions supported on a finite metric space and show consistency of the (naive) bootstrap. In particular, we prove that the empirical regularized transport plan itself asymptotically follows a Gaussian law. The theory includes the Boltzmann-Shannon entropy regularization and hence a limit law for the widely applied Sinkhorn divergence. Our approach is based on an application of the implicit function theorem to necessary and sufficient optimality conditions for the regularized transport problem. The asymptotic results are investigated in Monte Carlo simulations. We further discuss computational and statistical applications, e.g. confidence bands for colocalization analysis of protein interaction networks based on regularized optimal transport.