Minimax Optimal Estimation of Transport-Growth Pairs in Unbalanced Optimal Transport
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| Format: | Preprint |
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2026
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| _version_ | 1866914547483279360 |
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| author | Ponnoprat, Donlapark Isobe, Noboru Imaizumi, Masaaki |
| author_facet | Ponnoprat, Donlapark Isobe, Noboru Imaizumi, Masaaki |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_08705 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimax Optimal Estimation of Transport-Growth Pairs in Unbalanced Optimal Transport Ponnoprat, Donlapark Isobe, Noboru Imaizumi, Masaaki Statistics Theory Probability Methodology Machine Learning 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. |
| title | Minimax Optimal Estimation of Transport-Growth Pairs in Unbalanced Optimal Transport |
| topic | Statistics Theory Probability Methodology Machine Learning |
| url | https://arxiv.org/abs/2605.08705 |