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

• URL: http://arxiv.org/abs/2605.08705v1
• Date: Sat, 09 May 2026 05:39:47 GMT
• Title: Minimax Optimal Estimation of Transport-Growth Pairs in Unbalanced Optimal Transport
• Abstract summary: We study unbalanced transport with quadratic cost and Kullback-Leibler marginal penalties.<n>We argue that the natural population target is not a map alone, but a transport-growth pair.<n>We develop two estimators for the transport-growth pairs under several setups.
• Score: 6.281229317487582
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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