Unbalanced Kantorovich-Rubinstein distance and barycenter for finitely supported measures: A statistical perspective

We propose and investigate several statistical models and corresponding sampling schemes for data analysis based on unbalanced optimal transport (UOT) between finitely supported measures. Specifically, we analyse Kantorovich-Rubinstein (KR) distances with penalty parameter $C>0$. The main result provides non-asymptotic bounds on the expected error for the empirical KR distance as well as for its barycenters. The impact of the penalty parameter $C$ is studied in detail. Our approach justifies randomised computational schemes for UOT which can be used for fast approximate computations in combination with any exact solver. Using synthetic and real datasets, we empirically analyse the behaviour of the expected errors in simulation studies and illustrate the validity of our theoretical bounds.