🤖 AI Summary
Maximum likelihood estimation (MLE) for finite point processes—random point configurations on bounded domains—is computationally prohibitive, while existing score-matching methods lack rigorous theoretical foundations. Method: We establish the first score-matching framework grounded in the Janossy measure and propose an autoregressive weighted score-matching estimator. Theoretically, we expose an intrinsic identifiability limitation of nonparametric score matching—its inability to uniquely recover the true distribution—and introduce a survival-classification augmentation strategy to construct a differentiable training objective that avoids both integration and normalization constants. Our approach unifies Janossy measures, weighted score matching, autoregressive modeling, and survival loss, ensuring statistical consistency while drastically improving computational efficiency. Results: Experiments on synthetic and real spatiotemporal point process data demonstrate that our method accurately recovers intensity functions with accuracy comparable to MLE, while achieving significantly faster training.
📝 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.