Statistical unbalanced optimal transport

Music City Center 

Statistical optimal transport has become an emerging topic for the analysis of complex and geometric data. A fundamental assumption is that data are drawn randomly according to probability distributions. This, however, is often challenged in applications where modifications of optimal transport (unbalanced optimal transport, UOT) are successfully applied to situations when the underlying data do not come from a probability measure. This hinders a statistical analysis due to the lack of a valid random mechanism. In this talk we provide several statistical models where UOT becomes meaningful and develop first statistical theory for it. Specifically, we analyze extensions of the Kantorovich-Rubinstein (KR) transport for finitely supported measures. The KR transport depends on a penalty which serves as a relaxation from finding true couplings between the marginal measures. The main result is a non-asymptotic bound on the expected error for the empirical KR distance as well as for its barycenters. Depending on the penalty we find phase transitions, in analogue to the unbalanced case. Our approach justifies simple randomized 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 analyze the empirical UOT in simulation studies and investigate the validity of our theoretical bounds. Finally, UOT based inference is applied to protein colocalization in cell biology.

Kantorovich-Rubinstein transport

protein colocalization