From Jumps to Signatures: a Generative Method for Temporal Point Processes

2026-07-07

AI Summary

Existing methods struggle to apply path signature techniques to temporal point processes (TPPs) with jump discontinuities and lack sequence-level generative objectives and principled measures for distributional discrepancy. This work proposes a unified path-based framework that maps discrete event sequences to bounded-variation continuous paths via inter-arrival time embeddings, thereby extending rough path signatures to TPPs for the first time and introducing sigTPP—the first signature-based generative model for TPPs. The approach incorporates a path-level training objective and three theoretically grounded metrics for distributional divergence, overcoming the limitations of conventional event-wise optimization. Experiments demonstrate that sigTPP achieves the best average performance across eight evaluation metrics on both synthetic and real-world datasets, outperforming or matching the strongest baseline in 64% of dataset–metric combinations, with an average relative improvement of at least 19%.

Abstract

Rough path signatures are a universal feature map for continuous paths and, via the expected signature, characterise path distributions. These guarantees do not directly extend to cadlag paths of Temporal Point Processes (TPPs), limiting the use of signature methods for event sequences. Furthermore, neural TPP models, including recent generative approaches, optimise per-event objectives with no global sequence-level loss, while evaluation of variable-length event sequences lacks distributional discrepancy measures. This paper proposes a common pathwise framework for addressing these limitations. We introduce the interarrival embedding, a stable, injective lift from jump paths to continuous paths of bounded variation, extending signature methods to discrete event sequences. Our theoretical contributions give rise to sigTPP, the first signature-based generative model for TPPs, trained using a path-level loss on complete trajectories. We further analyse the space of counting paths and derive three distributional discrepancies, providing mathematically justified tools for evaluating generative TPP models. Across synthetic and real-world datasets, sigTPP achieves the best average rank based on eight complementary metrics, outperforms or is within a standard error of the strongest baseline in 64% of the dataset-metric pairs, and according to a relative score, improves against every baseline by at least 19% on average.

Problem

• Temporal Point Processes
• Rough Path Signatures
• Event Sequences
• Distributional Discrepancy
• Generative Modeling

Innovation

• signature
• temporal point processes
• generative modeling
• interarrival embedding
• distributional discrepancy