# Nonparametric estimation of Hawkes processes with RKHSs

arXiv: https://arxiv.org/abs/2411.00621 · HTML: https://mlsanity.com/paper/2411.00621/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 80 references that are in the map.
- First posted on arXiv 1 Nov 2024.

## Builds on (most cited first, 25 of 25)

- [Transformer Hawkes Process](https://mlsanity.com/paper/2002.09291/) (2020)
- [Intensity-Free Learning of Temporal Point Processes](https://mlsanity.com/paper/1909.12127/) (2019)
- [Transformer Embeddings of Irregularly Spaced Events and Their Participants](https://mlsanity.com/paper/2201.00044/) (2021)
- [Learning Granger Causality for Hawkes Processes](https://mlsanity.com/paper/1602.04511/) (2016)
- [Discovering Latent Network Structure in Point Process Data](https://mlsanity.com/paper/1402.0914/) (2014)
- [Variational Inference for Gaussian Process Modulated Poisson Processes](https://mlsanity.com/paper/1411.0254/) (2014)
- [Random Fourier Features for Operator-Valued Kernels](https://mlsanity.com/paper/1605.02536/) (2016)
- [Sparse and low-rank multivariate Hawkes processes](https://mlsanity.com/paper/1501.00725/) (2015)
- [Efficient Non-parametric Bayesian Hawkes Processes](https://mlsanity.com/paper/1810.03730/) (2018)
- [Poisson intensity estimation with reproducing kernels](https://mlsanity.com/paper/1610.08623/) (2016)
- [Distributed Inference for Linear Support Vector Machine](https://mlsanity.com/paper/1811.11922/) (2018)
- [Interpretable Transformer Hawkes Processes: Unveiling Complex Interactions in Social Networks](https://mlsanity.com/paper/2405.16059/) (2024)
- [Scalable Bayesian Inference for Excitatory Point Process Networks](https://mlsanity.com/paper/1507.03228/) (2015)
- [Learning with Smooth Hinge Losses](https://mlsanity.com/paper/2103.00233/) (2021)
- [Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity](https://mlsanity.com/paper/2205.04107/) (2022)
- [Interacting Hawkes processes with multiplicative inhibition](https://mlsanity.com/paper/2105.10597/) (2021)
- [Learning Granger Causality from Instance-wise Self-attentive Hawkes Processes](https://mlsanity.com/paper/2402.03726/) (2024)
- [Scalable Nonparametric Bayesian Inference on Point Processes with Gaussian Processes](https://mlsanity.com/paper/1410.6834/) (2014)
- [FaDIn: Fast Discretized Inference for Hawkes Processes with General Parametric Kernels](https://mlsanity.com/paper/2210.04635/) (2022)
- [Shallow Neural Hawkes: Non-parametric kernel estimation for Hawkes processes](https://mlsanity.com/paper/2006.02460/) (2020)
- [Flexible Parametric Inference for Space-Time Hawkes Processes](https://mlsanity.com/paper/2406.06849/) (2024)
- [Estimating Product Cannibalisation in Wholesale using Multivariate Hawkes Processes with Inhibition](https://mlsanity.com/paper/2201.05009/) (2022)
- [A flexible, random histogram kernel for discrete-time Hawkes processes](https://mlsanity.com/paper/2208.02921/) (2022)
- [Scalable Inference for Nonparametric Hawkes Process Using P'olya-Gamma Augmentation](https://mlsanity.com/paper/1910.13052/) (2019)
- [Non-Parametric Estimation of Multi-dimensional Marked Hawkes Processes](https://mlsanity.com/paper/2402.04740/) (2024)

## Built on by (most cited first, 2 of 2)

- [Advances in Temporal Point Processes: Bayesian, Neural, and LLM Approaches](https://mlsanity.com/paper/2501.14291/) (2025)
- [A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization](https://mlsanity.com/paper/2510.08916/) (2025)

## Abstract (arXiv)

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.