Nonparametric estimation of Hawkes processes with RKHSs

Anna Bonnet, Maxime Sangnier
Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, PMLR 258:3574-3582, 2025.

Abstract

This paper addresses nonparametric estimation of nonlinear multivariate Hawkes processes, where the interaction functions are assumed to lie in a reproducing kernel Hilbert space (RKHS). Motivated by applications in neuroscience, the model allows complex interaction functions, in order to express exciting and inhibiting effects, but also a combination of both (which is particularly interesting to model the refractory period of neurons), and considers in return that conditional intensities are rectified by the ReLU function. The latter feature incurs several methodological challenges, for which workarounds are proposed in this paper. In particular, it is shown that a representer theorem can be obtained for approximated versions of the log-likelihood and the least-squares criteria. Based on it, we propose an estimation method, that relies on two common approximations (of the ReLU function and of the integral operator). We provide a bound that controls the impact of these approximations. Numerical results on synthetic data confirm this fact as well as the good asymptotic behavior of the proposed estimator. It also shows that our method achieves a better performance compared to related nonparametric estimation techniques and suits neuronal applications.

Cite this Paper

BibTeX
@InProceedings{pmlr-v258-bonnet25a, title = {Nonparametric estimation of Hawkes processes with RKHSs}, author = {Bonnet, Anna and Sangnier, Maxime}, booktitle = {Proceedings of The 28th International Conference on Artificial Intelligence and Statistics}, pages = {3574--3582}, year = {2025}, editor = {Li, Yingzhen and Mandt, Stephan and Agrawal, Shipra and Khan, Emtiyaz}, volume = {258}, series = {Proceedings of Machine Learning Research}, month = {03--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v258/main/assets/bonnet25a/bonnet25a.pdf}, url = {https://proceedings.mlr.press/v258/bonnet25a.html}, abstract = {This paper addresses nonparametric estimation of nonlinear multivariate Hawkes processes, where the interaction functions are assumed to lie in a reproducing kernel Hilbert space (RKHS). Motivated by applications in neuroscience, the model allows complex interaction functions, in order to express exciting and inhibiting effects, but also a combination of both (which is particularly interesting to model the refractory period of neurons), and considers in return that conditional intensities are rectified by the ReLU function. The latter feature incurs several methodological challenges, for which workarounds are proposed in this paper. In particular, it is shown that a representer theorem can be obtained for approximated versions of the log-likelihood and the least-squares criteria. Based on it, we propose an estimation method, that relies on two common approximations (of the ReLU function and of the integral operator). We provide a bound that controls the impact of these approximations. Numerical results on synthetic data confirm this fact as well as the good asymptotic behavior of the proposed estimator. It also shows that our method achieves a better performance compared to related nonparametric estimation techniques and suits neuronal applications.} }
Endnote
%0 Conference Paper %T Nonparametric estimation of Hawkes processes with RKHSs %A Anna Bonnet %A Maxime Sangnier %B Proceedings of The 28th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2025 %E Yingzhen Li %E Stephan Mandt %E Shipra Agrawal %E Emtiyaz Khan %F pmlr-v258-bonnet25a %I PMLR %P 3574--3582 %U https://proceedings.mlr.press/v258/bonnet25a.html %V 258 %X This paper addresses nonparametric estimation of nonlinear multivariate Hawkes processes, where the interaction functions are assumed to lie in a reproducing kernel Hilbert space (RKHS). Motivated by applications in neuroscience, the model allows complex interaction functions, in order to express exciting and inhibiting effects, but also a combination of both (which is particularly interesting to model the refractory period of neurons), and considers in return that conditional intensities are rectified by the ReLU function. The latter feature incurs several methodological challenges, for which workarounds are proposed in this paper. In particular, it is shown that a representer theorem can be obtained for approximated versions of the log-likelihood and the least-squares criteria. Based on it, we propose an estimation method, that relies on two common approximations (of the ReLU function and of the integral operator). We provide a bound that controls the impact of these approximations. Numerical results on synthetic data confirm this fact as well as the good asymptotic behavior of the proposed estimator. It also shows that our method achieves a better performance compared to related nonparametric estimation techniques and suits neuronal applications.
APA
Bonnet, A. & Sangnier, M.. (2025). Nonparametric estimation of Hawkes processes with RKHSs. Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 258:3574-3582 Available from https://proceedings.mlr.press/v258/bonnet25a.html.

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