  High-dimensional autoregressive point processes model how current events
trigger or inhibit future events, such as activity by one member of a social
network can affect the future activity of his or her neighbors. While past work
has focused on estimating the underlying network structure based solely on the
times at which events occur on each node of the network, this paper examines
the more nuanced problem of estimating context-dependent networks that reflect
how features associated with an event (such as the content of a social media
post) modulate the strength of influences among nodes. Specifically, we
leverage ideas from compositional time series and regularization methods in
machine learning to conduct network estimation for high-dimensional marked
point processes. Two models and corresponding estimators are considered in
detail: an autoregressive multinomial model suited to categorical marks and a
logistic-normal model suited to marks with mixed membership in different
categories. Importantly, the logistic-normal model leads to a convex negative
log-likelihood objective and captures dependence across categories. We provide
theoretical guarantees for both estimators, which we validate by simulations
and a synthetic data-generating model. We further validate our methods through
two real data examples and demonstrate the advantages and disadvantages of both
approaches.
