[Paper Review] Fused Gromov-Wasserstein Alignment for Hawkes Processes

Dixin Luo, Hongteng Xu+1 more|arXiv (Cornell University)|Oct 4, 2019

Point processes and geometric inequalities10 references4 citations

TL;DR

This paper proposes a fused Gromov-Wasserstein alignment (FGWA) method to jointly learn Hawkes processes and align event types across different temporal event sequences by minimizing a regularized likelihood objective. The method combines Wasserstein discrepancy on base intensities and Gromov-Wasserstein discrepancy on infectivity matrices via optimal transport, achieving superior alignment accuracy and higher certainty in correspondence learning on synthetic and real-world data.

ABSTRACT

We propose a novel fused Gromov-Wasserstein alignment method to jointly learn the Hawkes processes in different event spaces, and align their event types. Given two Hawkes processes, we use fused Gromov-Wasserstein discrepancy to measure their dissimilarity, which considers both the Wasserstein discrepancy based on their base intensities and the Gromov-Wasserstein discrepancy based on their infectivity matrices. Accordingly, the learned optimal transport reflects the correspondence between the event types of these two Hawkes processes. The Hawkes processes and their optimal transport are learned jointly via maximum likelihood estimation, with a fused Gromov-Wasserstein regularizer. Experimental results show that the proposed method works well on synthetic and real-world data.

Motivation & Objective

• To address the challenge of aligning event types across different Hawkes processes in distinct event spaces.
• To jointly learn Hawkes process parameters and optimal transport maps that reflect cross-process event type correspondences.
• To improve alignment accuracy and correspondence certainty by integrating both base intensity and infectivity matrix discrepancies.
• To develop a scalable, differentiable framework for cross-domain sequential behavior alignment using optimal transport.

Proposed method

• The method models event sequences as Hawkes processes parameterized by base intensity vectors and infectivity matrices.
• It introduces a fused Gromov-Wasserstein discrepancy that combines Wasserstein distance on base intensities and Gromov-Wasserstein distance on infectivity matrices.
• The optimal transport map is learned via maximum likelihood estimation with a fused Gromov-Wasserstein regularizer to align event types across domains.
• The regularizer encourages correspondence learning by minimizing the combined discrepancy between source and target processes.
• The framework jointly optimizes Hawkes process parameters and the transport map using stochastic gradient descent.
• The method uses Ogata’s thinning algorithm to simulate event sequences and evaluates performance via alignment accuracy and entropy of the transport matrix.

Experimental results

Research questions

• RQ1 Can fused Gromov-Wasserstein discrepancy effectively align event types across Hawkes processes with different event spaces?
• RQ2 How does combining base intensity and infectivity matrix discrepancies improve alignment certainty and accuracy?
• RQ3 Does the proposed FGWA method outperform existing methods like empirical matching, HP-WD, and HP-GWD in event type alignment?
• RQ4 To what extent does the optimal transport matrix learned by FGWA reflect a bijective and certain correspondence between event types?
• RQ5 How scalable and robust is the FGWA method on real-world datasets with non-bijective correspondences?

Key findings

• On synthetic data with C=10, FGWA achieved 69% top-1 alignment accuracy, outperforming HP-GWD (49%) and HP-WD (43%).
• For C=50, FGWA achieved 22% top-1 accuracy, significantly outperforming HP-GWD (18%) and HP-WD (19%).
• On C=100 synthetic data, FGWA maintained 11% top-1 accuracy, surpassing HP-GWD (6%) and HP-WD (6%).
• In the MIMIC-III dataset, FGWA achieved 46.4% top-K (K=5) alignment accuracy, outperforming HP-GWD (31.4%) and HP-WD (33.2%).
• On the MC3 dataset, FGWA achieved 25.3% top-K (K=50) accuracy, exceeding HP-GWD (12.9%) and HP-WD (17.7%).
• The entropy of the learned transport matrix in FGWA was consistently lower than other methods, indicating higher certainty in the predicted correspondences.