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

arXiv: https://arxiv.org/abs/2605.08705 · HTML: https://mlsanity.com/paper/2605.08705/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 42 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 3 of 3)

- [Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implications](https://mlsanity.com/paper/2502.12326/) (2025)
- [Minimax Rates of Estimation for Optimal Transport Map between Infinite-Dimensional Spaces](https://mlsanity.com/paper/2505.13570/) (2025)
- [Sharp Convergence Rates of Empirical Unbalanced Optimal Transport for Spatio-Temporal Point Processes](https://mlsanity.com/paper/2509.04225/) (2025)

## Built on by (most cited first, 1 of 1)

- [A Neural JKO Scheme for Hellinger-Kantorovich Gradient Flows via Monge-Growth Pairs](https://mlsanity.com/paper/2610.07602/) (2026)

## Abstract (arXiv)

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.