  Despite the fundamental nature of the inhomogeneous Poisson process in the
theory and application of stochastic processes, and its attractive
generalizations (e.g. Cox process), few tractable nonparametric modeling
approaches of intensity functions exist, especially when observed points lie in
a high-dimensional space. In this paper we develop a new, computationally
tractable Reproducing Kernel Hilbert Space (RKHS) formulation for the
inhomogeneous Poisson process. We model the square root of the intensity as an
RKHS function. Whereas RKHS models used in supervised learning rely on the
so-called representer theorem, the form of the inhomogeneous Poisson process
likelihood means that the representer theorem does not apply. However, we prove
that the representer theorem does hold in an appropriately transformed RKHS,
guaranteeing that the optimization of the penalized likelihood can be cast as a
tractable finite-dimensional problem. The resulting approach is simple to
implement, and readily scales to high dimensions and large-scale datasets.
