Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation

Isaias Banales; Arturo Jaramillo

stat.MLOct 14, 2026

Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation

Authors: Isaias Banales, Arturo Jaramillo

Organizations: Facultad de Ciencias, Universidad Aut´onoma de M´exico · Department of Probability and Statistics, Centro de Investigaci´on en Matem´aticas A.C., Guanajuato, M´exico

Abstract

We consider the problem of estimating the background intensity of a Hawkes process when the triggering mechanism is known, treating the background as an unknown finite positive measure with a regular density. We propose a likelihood-based variational approach based on the Wasserstein-Fisher-Rao geometry. In this setting, the relevant first variations are naturally distribution-valued and therefore fall outside the classical gradient-flow formulation. We develop a distributional framework for the resulting dynamics and give them a finite-dimensional meaning through projections onto statistical manifolds. We identify the true background as the unique zero of a limiting generalized gradient and, under suitable stability, compactness, integrability and approximation conditions, establish a conditional consistency theorem for measurable estimators satisfying the empirical projected equations on events whose probabilities tend to one. Finally, we specialize the construction to Gaussian-mixture manifolds and obtain an explicit finite-dimensional scheme.

Figures & tables

$\sigma^{2}$	AIC	BIC	Log-Likelihood	Components ( $k$ )	Parameters
0.10	171498.49	172057.39	-85626.24	41	123
0.35	3246.64	3587.44	-1548.32	25	75
0.60	1471.30	1703.04	-684.65	17	51
0.85	1260.48	1437.70	-591.24	13	39
1.10	1194.08	1303.13	-573.04	8	24
1.35	1227.60	1309.39	-595.80	6	18

Table 1. Model evaluation metrics over a grid search of

\sigma^{2}

bandwidth values for the Poisson process case.

Panel of Figure 1 — Figure 1. Comparison of the process intensity function. The red dots represent the $m_{i}$ values of the mixture elements

$\sigma^{2}$	AIC	BIC	Log-Likelihood	Components ( $k$ )	Parameters
0.10	30448.32	30626.05	-15186.16	11	38
0.20	10954.69	11076.30	-5451.35	7	26
0.30	8497.65	8619.26	-4222.83	7	26
0.40	7957.68	8065.26	-3955.84	6	23
0.50	7852.39	7959.96	-3903.19	6	23
0.60	7851.05	7958.63	-3902.53	6	23

Table 2. Model evaluation metrics over a grid search of

\sigma^{2}

bandwidth values for the Hawkes process case.

Panel of Figure 2 — Figure 2. Spatial distribution of seismic events in Southern Mexico from the USGS catalog (2000–2016) and the estimated ETAS background intensity function. The Middle America Trench is shown in pink, and the red dots represent the mixture element centers $m_{i}$ , ( $i=1,2,…,6$ ).

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

\bm{\Theta}_{\text{bg}}=(\mathbf{m}_{1},\dots,\mathbf{m}_{k},\alpha_{1},\dots,\alpha_{k})

Appendix

Algorithm 1 WFR-Hawkes Algorithm (Spatial Background & Kernel Estimation)

Hawkes Processes

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Massively Parallel Exact Inference for Hawkes Processes

Ahmer Raza, Hudson Smith

Clemson University

Hawkes Processes Temporal Point Processes GPU Acceleration

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Multivariate Hawkes processes are a widely used class of self-exciting point processes, but maximum likelihood estimation naively scales as

O(N^2)

in the number of events. The canonical linear exponential Hawkes process admits a faster

O(N)

recurrence, but prior work evaluates this recurrence sequentially, without exploiting parallelization on modern GPUs. We show that the Hawkes process intensity can be expressed as a product of sparse transition matrices admitting a linear-time associative multiply, enabling computation via a parallel prefix scan. This yields a massively parallelizable algorithm for estimation of linear exponential Hawkes processes. Our method reduces the computational complexity to approximately

O(N/P)

with

P

parallel processors, and naturally yields a batching scheme to maintain constant memory usage, avoiding GPU memory constraints. Importantly, it computes the exact likelihood without any additional assumptions or approximations, preserving the simplicity and interpretability of the model. We demonstrate orders-of-magnitude speedups on simulated and real datasets, scaling to thousands of nodes and tens of millions of events, substantially beyond scales reported in prior work. We provide an open-source PyTorch library implementing our optimizations.

Abstract

Hawkes Processes

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