# A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization

arXiv: https://arxiv.org/abs/2510.08916 · HTML: https://mlsanity.com/paper/2510.08916/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 48 references that are in the map.
- First posted on arXiv 10 Oct 2025.

## Builds on (most cited first, 9 of 9)

- [Transformer Hawkes Process](https://mlsanity.com/paper/2002.09291/) (2020)
- [Learning Granger Causality for Hawkes Processes](https://mlsanity.com/paper/1602.04511/) (2016)
- [Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels](https://mlsanity.com/paper/1412.8293/) (2014)
- [Sparse and low-rank multivariate Hawkes processes](https://mlsanity.com/paper/1501.00725/) (2015)
- [Poisson intensity estimation with reproducing kernels](https://mlsanity.com/paper/1610.08623/) (2016)
- [Latent Network Structure Learning from High Dimensional Multivariate Point Processes](https://mlsanity.com/paper/2004.03569/) (2020)
- [Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity](https://mlsanity.com/paper/2205.04107/) (2022)
- [Mean-field inference of Hawkes point processes](https://mlsanity.com/paper/1511.01512/) (2015)
- [Nonparametric estimation of Hawkes processes with RKHSs](https://mlsanity.com/paper/2411.00621/) (2024)

## Abstract (arXiv)

The representer theorem is a cornerstone of kernel methods, which aim to estimate latent functions in reproducing kernel Hilbert spaces (RKHSs) in a nonparametric manner. Its significance lies in converting inherently infinite-dimensional optimization problems into finite-dimensional ones over dual coefficients, thereby enabling practical and computationally tractable algorithms. In this paper, we address the problem of estimating the latent triggering kernels--functions that encode the interaction structure between events--for linear multivariate Hawkes processes based on observed event sequences within an RKHS framework. We show that, under the principle of penalized least squares minimization, a novel form of representer theorem emerges: a family of transformed kernels can be defined via a system of simultaneous integral equations, and the optimal estimator of each triggering kernel is expressed as a linear combination of these transformed kernels evaluated at the data points. Remarkably, the dual coefficients are all analytically fixed to unity, obviating the need to solve a costly optimization problem to obtain the dual coefficients. This leads to a highly efficient estimator capable of handling large-scale data more effectively than conventional nonparametric approaches. Empirical evaluations on synthetic datasets reveal that the proposed method attains competitive predictive accuracy while substantially improving computational efficiency over existing state-of-the-art kernel method-based estimators.