Poisson intensity estimation with reproducing kernels

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 in high dimensional settings. 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. The modeling challenge is that the usual representer theorem arguments no longer apply due to the form of the inhomogeneous Poisson process likelihood. 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.