Unbalanced Kantorovich-Rubinstein distance, plan, and barycenter on finite spaces: A statistical perspective

TL;DR

This paper investigates the statistical properties of plug-in estimators for unbalanced optimal transport measures, providing non-asymptotic error bounds and computational schemes, validated through synthetic and real data experiments.

Contribution

It offers the first non-asymptotic bounds for empirical unbalanced Kantorovich-Rubinstein distances, plans, and barycenters, and introduces randomized algorithms for efficient approximation.

Findings

Non-asymptotic error bounds are established for empirical estimates.

Randomized computational schemes are justified for fast approximation.

Empirical results validate theoretical bounds using synthetic and real datasets.

Abstract

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.