Poisson intensity estimation with reproducing kernels

TL;DR

This paper introduces a new RKHS-based method for nonparametric estimation of Poisson process intensities, enabling scalable and tractable modeling in high-dimensional spaces.

Contribution

It develops a novel RKHS formulation for Poisson intensities using the square root transformation, overcoming representer theorem limitations for efficient optimization.

Findings

Method is computationally tractable and scalable.

Applicable to high-dimensional and large datasets.

Provides a new theoretical foundation for Poisson intensity estimation.

Abstract

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.