# Sharp Convergence Rates of Empirical Unbalanced Optimal Transport for Spatio-Temporal Point Processes

arXiv: https://arxiv.org/abs/2509.04225 · HTML: https://mlsanity.com/paper/2509.04225/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Sep 2025.

## Built on by (most cited first, 2 of 2)

- [Minimax Optimal Estimation of Transport-Growth Pairs in Unbalanced Optimal Transport](https://mlsanity.com/paper/2605.08705/) (2026)
- [Quantifying the noise sensitivity of the Wasserstein metric for images](https://mlsanity.com/paper/2510.01015/) (2025)

## Abstract (arXiv)

We statistically analyze empirical plug-in estimators for unbalanced optimal transport (UOT) formalisms, focusing on the Kantorovich-Rubinstein distance, between general intensity measures based on observations from spatio-temporal point processes. Specifically, we model the observations by two weakly time-stationary point processes with spatial intensity measures μ and nu over the expanding window (0,t] as t increases to infinity, and establish sharp convergence rates of the empirical UOT in terms of the intrinsic dimensions of the measures. We assume a sub-quadratic temporal growth condition of the variance of the process, which allows for a wide range of temporal dependencies. As the growth approaches quadratic, the convergence rate becomes slower. This variance assumption is related to the time-reduced factorial covariance measure, and we exemplify its validity for various point processes, including the Poisson cluster, Hawkes, Neyman-Scott, and log-Gaussian Cox processes. Complementary to our upper bounds, we also derive matching lower bounds for various spatio-temporal point processes of interest and establish near minimax rate optimality of the empirical Kantorovich-Rubinstein distance.