Nonparametric estimation of Hawkes processes with RKHSs

TL;DR

The paper tackles offline, nonparametric estimation of nonlinear multivariate Hawkes processes by embedding interaction kernels in a reproducing kernel Hilbert space and rectifying intensities with a ReLU link. It derives representer theorems for approximated likelihood and least-squares criteria and introduces a practical kernelized estimator based on three linked approximations: RKHS lineages for $h_{j\ell}$, a Riemann-sum approximation of integrals, and a softplus surrogate for ReLU, with explicit error bounds showing $O(1/M)+O(1/\omega)$ control. Theoretical guarantees and numerical experiments on synthetic and neuronal spike data demonstrate that the proposed RKHS method robustly captures both excitatory and inhibitory interactions, including refractory periods, and often outperforms exponential and Bernstein-based nonparametric methods in terms of kernel recovery and log-likelihood. The work provides a scalable, flexible framework for modeling complex neuronal interactions with potential extensions to sparse, spatiotemporal, and operator-valued kernels, broadening the toolbox for neuroscience data analysis.

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.

Figures (8)

• Figure 1: Example of estimation of the triggering kernels (blue) of a Hawkes process with the exponential model (orange) and the proposed method (green).
• Figure 2: Example of estimations with horizon $T = 2000$.
• Figure 3: Approximation error of true kernels with respect to the hyperparameters $\omega$ and $M$.
• Figure 4: Approximation error of true kernels with respect to the horizon $T$.
• Figure 5: Test log-likelihood with respect to the horizon $T$.

Theorems & Definitions (12)

• Proposition 1: Representer theorem for discretized least-squares estimation of linear Hawkes processes
• Proposition 2: Representer theorem for approximated maximum likelihood estimation
• Proposition 3: Representer theorem for approximate least-squares estimation
• Proposition 4: Approximation quality for maximum likelihood estimation
• Proposition 5: Approximation quality for least-squares estimation
• Remark D.1
• Lemma 6
• proof
• Lemma 7
• proof