  We present the Additive Poisson Process (APP), a novel framework that can
model the higher-order interaction effects of the intensity functions in
stochastic processes using lower dimensional projections. Our model combines
the techniques in information geometry to model higher-order interactions on a
statistical manifold and in generalized additive models to use
lower-dimensional projections to overcome the effects from the curse of
dimensionality. Our approach solves a convex optimization problem by minimizing
the KL divergence from a sample distribution in lower dimensional projections
to the distribution modeled by an intensity function in the stochastic process.
Our empirical results show that our model is able to use samples observed in
the lower dimensional space to estimate the higher-order intensity function
with extremely sparse observations.
