Minimax Optimal Estimation of Transport-Growth Pairs in Unbalanced Optimal Transport

TL;DR

This paper develops minimax optimal estimators for transport-growth pairs in unbalanced optimal transport, providing statistical guarantees and a theoretical foundation for Monge-type estimation.

Contribution

It introduces two estimators for transport-growth pairs in unbalanced OT and proves their minimax optimality with a novel stability reduction technique.

Findings

Estimator achieves minimax optimal rate.

Derived a matching lower bound for minimax risk.

Provided a stability reduction technique for UOT.

Abstract

Unbalanced optimal transport (UOT) extends classical optimal transport to measures with different total masses, but statistical guarantees for Monge-type estimation remain limited. We study unbalanced transport with quadratic cost and Kullback-Leibler marginal penalties and argue that the natural population target is not a map alone, but a transport-growth pair. Consequently, we develop two estimators for the transport-growth pairs under several setups: an optimal transport plan-based estimator for a general case, and a kernel-based estimator for a case with smooth densities. We also show that an error of the estimator achieves the minimax optimal rate by deriving a matching lower bound of the minimax risk. Our main technical contribution is a value-based stability reduction that converts perturbations of the UOT objective into transport and growth risks through a UOT gap condition. These results provide a statistical foundation for Monge-type estimation in unbalanced optimal transport.