Score Matching for Estimating Finite Point Processes

TL;DR

This paper develops a rigorous framework for score matching in finite point processes, introduces an autoregressive estimator, and proposes a survival-classification augmentation to improve model training, demonstrating effectiveness on real datasets.

Contribution

It provides the first formal analysis of score matching for finite point processes and introduces an augmentation for nonparametric models to ensure proper normalization.

Findings

Accurately recovers intensities in synthetic and real data.

Achieves performance comparable to MLE with improved efficiency.

Addresses normalization issues in nonparametric point process models.

Abstract

Score matching estimators have garnered significant attention in recent years because they eliminate the need to compute normalizing constants, thereby mitigating the computational challenges associated with maximum likelihood estimation (MLE).While several studies have proposed score matching estimators for point processes, this work highlights the limitations of these existing methods, which stem primarily from the lack of a mathematically rigorous analysis of how score matching behaves on finite point processes -- special random configurations on bounded spaces where many of the usual assumptions and properties of score matching no longer hold. To this end, we develop a formal framework for score matching on finite point processes via Janossy measures and, within this framework, introduce an (autoregressive) weighted score-matching estimator, whose statistical properties we analyze in classical parametric settings. For general nonparametric (e.g., deep) point process models, we show that score matching alone does not uniquely identify the ground-truth distribution due to subtle normalization issues, and we propose a simple survival-classification augmentation that yields a complete, integration-free training objective for any intensity-based point process model for spatio-temporal case. Experiments on synthetic and real-world temporal and spatio-temporal datasets, demonstrate that our method accurately recovers intensities and achieves performance comparable to MLE with better efficiency.