A Representer Theorem for Coarse-Grained Hawkes Processes

Hawkes processes are a standard framework for modeling event sequences and estimating triggering kernels that characterize inter-event influences from event data. In many practical settings, however, observations are limited to event counts within fixed time bins or windows rather than exact event timestamps. This challenging observation regime has motivated growing interest in estimating triggering kernels on aggregated event data. Among the approaches developed for this setting, the recently proposed *coarse-grained Hawkes process* (CGHP) offers a parametric framework that combines computational efficiency with robustness to temporal aggregation. In this paper, we propose a nonparametric extension of CGHP within a reproducing kernel Hilbert space (RKHS) framework, enabling flexible estimation of triggering kernels from aggregated observations. By exploiting the mathematical structure of CGHP, we establish a representer theorem, a cornerstone of kernel methods, that reduces the infinite-dimensional estimation problem to a finite-dimensional one, providing the foundation for an efficient estimation algorithm. To the best of our knowledge, this is the first representer theorem for a Hawkes process model designed for aggregated event data. Experiments on synthetic and real-world datasets demonstrate that the proposed method achieves higher estimation accuracy than the state-of-the-art nonparametric approach, with particularly pronounced improvements under coarse temporal aggregation.