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.

翻译：表示定理是核方法的基石，其目标是以非参数方式估计再生核希尔伯特空间中的隐函数。该定理的重要性在于将本质上是无限维的优化问题转化为对偶系数上的有限维问题，从而实现实用且计算上可处理的算法。本文在RKHS框架下，基于观测到的事件序列，解决了估计线性多元霍克斯过程的潜在触发核——编码事件间交互结构的函数——的问题。我们证明，在惩罚最小二乘最小化原则下，会出现一种新形式的表示定理：可以通过一个联立积分方程组定义一族变换核，并且每个触发核的最优估计量表示为这些变换核在数据点处取值的线性组合。值得注意的是，所有对偶系数都被解析地固定为1，从而无需通过求解昂贵的优化问题来获得对偶系数。这产生了一种高效的估计器，能够比传统的非参数方法更有效地处理大规模数据。在合成数据集上的实证评估表明，所提出的方法在达到有竞争力的预测精度的同时，相比现有最先进的基于核方法的估计器，计算效率得到了显著提升。