A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization

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.

💡 Research Summary

The paper tackles the non‑parametric estimation of triggering kernels in linear multivariate Hawkes processes by embedding the problem in a reproducing kernel Hilbert space (RKHS) and minimizing a penalized least‑squares (LS) loss. Classical kernel methods rely on a representer theorem that reduces an infinite‑dimensional optimization to a finite sum over data points, but they still require solving for dual coefficients whose number grows with the data size. In the context of point processes, the loss involves integrals of the intensity function, which makes the standard representer theorem inapplicable or computationally burdensome.

The authors propose a novel representer theorem tailored to the LS formulation for Hawkes processes. They show that the optimal estimator of each triggering kernel (g_{ij}) can be expressed as
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