Sharp Convergence Rates of Empirical Unbalanced Optimal Transport for Spatio-Temporal Point Processes
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914022477004800 |
|---|---|
| author | Struleva, Marina Hundrieser, Shayan Schuhmacher, Dominic Munk, Axel |
| author_facet | Struleva, Marina Hundrieser, Shayan Schuhmacher, Dominic Munk, Axel |
| contents | 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 $ν$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04225 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Convergence Rates of Empirical Unbalanced Optimal Transport for Spatio-Temporal Point Processes Struleva, Marina Hundrieser, Shayan Schuhmacher, Dominic Munk, Axel Statistics Theory Machine Learning primary 62G05, 62G07, 62R20, secondary: 60D05, 60G60 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 $ν$ 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. |
| title | Sharp Convergence Rates of Empirical Unbalanced Optimal Transport for Spatio-Temporal Point Processes |
| topic | Statistics Theory Machine Learning primary 62G05, 62G07, 62R20, secondary: 60D05, 60G60 |
| url | https://arxiv.org/abs/2509.04225 |