  In this paper, we address the problem of fitting multivariate Hawkes
processes to potentially large-scale data in a setting where series of events
are not only mutually-exciting but can also exhibit inhibitive patterns. We
focus on nonparametric learning and propose a novel algorithm called MEMIP
(Markovian Estimation of Mutually Interacting Processes) that makes use of
polynomial approximation theory and self-concordant analysis in order to learn
both triggering kernels and base intensities of events. Moreover, considering
that N historical observations are available, the algorithm performs
log-likelihood maximization in $O(N)$ operations, while the complexity of
non-Markovian methods is in $O(N^{2})$. Numerical experiments on simulated
data, as well as real-world data, show that our method enjoys improved
prediction performance when compared to state-of-the art methods like MMEL and
exponential kernels.
