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

2026-05-09

AI Summary

This work addresses the lack of statistical guarantees and the neglect of mass variation in existing Monge-type estimators for unbalanced optimal transport. The authors reformulate the objective as a transport–growth pair and develop estimators tailored to both general and smooth density settings, leveraging optimal transport plans and kernel methods, respectively. By introducing a value stability reduction technique and integrating a quadratic-cost model with KL marginal penalization, kernel density estimation, and minimax lower bound analysis, they establish—for the first time—that the proposed estimators achieve minimax-optimal convergence rates across multiple regimes. This provides a rigorous statistical foundation for Monge-type estimation in unbalanced optimal transport.

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.

Problem

• unbalanced optimal transport
• transport-growth pair
• Monge-type estimation
• minimax optimality
• statistical guarantees

Innovation

• unbalanced optimal transport
• transport-growth pair
• minimax optimal rate
• value-based stability reduction
• Monge-type estimation