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.

翻译：本文研究非线性多元霍克斯过程的非参数估计问题，其中交互函数被假定位于再生核希尔伯特空间（RKHS）。受神经科学应用的启发，该模型允许复杂的交互函数，既能表达兴奋性与抑制性效应，也能实现两者的组合（这对模拟神经元的不应期特别有意义），并相应假设条件强度通过ReLU函数进行整流。后一特性引发了若干方法论挑战，本文针对这些挑战提出了解决方案。特别地，我们证明了对于对数似然准则与最小二乘准则的近似形式，均可获得表示定理。基于此定理，我们提出了一种依赖于两种常见近似（ReLU函数近似与积分算子近似）的估计方法，并给出了控制这些近似误差的界。在合成数据上的数值实验结果验证了该界的有效性以及所提估计量的良好渐近性质。实验还表明，相较于相关非参数估计技术，我们的方法取得了更优的性能，且适用于神经元应用场景。