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

2025-09-04

AI Summary

This paper addresses the empirical estimation of unbalanced optimal transport (UOT) between generalized intensity measures derived from spatiotemporal point process observations, focusing on the statistical convergence of the Kantorovich–Rubinstein distance. Under an expanding observation window, we establish sharp convergence rates for the empirical UOT estimator, with explicit upper and lower bounds depending on the intrinsic dimension of the intensity measures—achieving near minimax optimality. Methodologically, under weak stationarity, subquadratic variance growth, and factorial cumulant reducibility, our analysis unifies key models including Poisson cluster, Hawkes, Neyman–Scott, and log-Gaussian Cox processes. To our knowledge, this is the first work to extend UOT statistical theory to spatiotemporal point processes exhibiting complex temporal dependence. The results provide rigorous convergence guarantees and a dimension-adaptive theoretical foundation for robust UOT applications to real-world spatiotemporal data.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Stochastic Optimization
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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.

Problem

• Estimating unbalanced optimal transport between intensity measures
• Analyzing convergence rates for spatio-temporal point processes
• Establishing minimax optimality of empirical Kantorovich-Rubinstein distance

Innovation

• Empirical plug-in estimators for unbalanced optimal transport
• Sharp convergence rates for spatio-temporal point processes
• Matching lower bounds establishing minimax optimality