A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization

• URL: http://arxiv.org/abs/2510.08916v1
• Date: Fri, 10 Oct 2025 02:00:56 GMT
• Title: A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization
• Authors: Hideaki Kim, Tomoharu Iwata,
• Abstract summary: The representer theorem is a cornerstone of kernel methods, which aim to estimate latent functions in reproducing kernel Hilbert spaces.<n>We show that a novel form of representer theorem emerges: a family of transformed kernels can be defined via a system of simultaneous integral equations.<n>Remarkably, the dual coefficients are all analytically fixed to unity, obviating the need to solve a costly optimization problem to obtain the dual coefficients.
• Score: 31.876688992403647
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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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