通过惩罚最小二乘法的Hawkes过程的代表定理
A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization
AI总结：
本文提出了一种基于惩罚最小二乘法的Hawkes过程代表定理，通过转换核和固定dual系数，提高了大规模数据处理效率和预测准确性。
摘要：
代表定理是核方法的核心，旨在以非参数方式估计 reproducing kernel Hilbert spaces (RKHSs) 中的潜在函数。其意义在于将本质上无限维的优化问题转换为有限维的 dual 系数问题，从而使得实际且计算上可行的算法成为可能。在本文中，我们基于观察到的事件序列，在 RKHS 框架内解决线性多元 Hawkes 过程中潜在触发核（编码事件之间相互作用结构的函数）的估计问题。我们证明，在惩罚最小二乘法原理下，一种新的代表定理形式出现：通过一组同时积分方程可以定义一组转换后的核，每个触发核的最佳估计器可以表示为这些转换后的核在数据点处的线性组合。值得注意的是，所有 dual 系数都被解析地固定为单位，从而避免了通过解决昂贵的优化问题来获得 dual 系数的需要。这导致了一个高效估计器，能够更有效地处理大规模数据，比传统非参数方法更有效。在合成数据集上的经验评估显示，所提出的方法在预测准确性方面具有竞争力，同时在现有最先进核方法估计器上显著提高了计算效率。
英文摘要：
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.
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