# Poisson intensity estimation with reproducing kernels

arXiv: https://arxiv.org/abs/1610.08623 · HTML: https://mlsanity.com/paper/1610.08623/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Oct 2016.

## Built on by (most cited first, 14 of 14)

- [Squared Neural Families: A New Class of Tractable Density Models](https://mlsanity.com/paper/2305.13552/) (2023)
- [Exact, Fast and Expressive Poisson Point Processes via Squared Neural Families](https://mlsanity.com/paper/2402.09608/) (2024)
- [Bayesian Optimization through Gaussian Cox Process Models for Spatio-temporal Data](https://mlsanity.com/paper/2401.14544/) (2024)
- [Sensing Cox Processes via Posterior Sampling and Positive Bases](https://mlsanity.com/paper/2110.11181/) (2021)
- [Advances in Temporal Point Processes: Bayesian, Neural, and LLM Approaches](https://mlsanity.com/paper/2501.14291/) (2025)
- [Kernelized Normalizing Constant Estimation: Bridging Bayesian Quadrature and Bayesian Optimization](https://mlsanity.com/paper/2401.05716/) (2024)
- [Nonstationary Sparse Spectral Permanental Process](https://mlsanity.com/paper/2410.03581/) (2024)
- [Nonparametric estimation of Hawkes processes with RKHSs](https://mlsanity.com/paper/2411.00621/) (2024)
- [Squared families: Searching beyond regular probability models](https://mlsanity.com/paper/2503.21128/) (2025)
- [Spatio-temporal extreme event modeling of terror insurgencies](https://mlsanity.com/paper/2110.08363/) (2021)
- [K2IE: Kernel Method-based Kernel Intensity Estimators for Inhomogeneous Poisson Processes](https://mlsanity.com/paper/2505.24704/) (2025)
- [A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization](https://mlsanity.com/paper/2510.08916/) (2025)
- [Non-Negative Matrix Factorization for Event Data](https://mlsanity.com/paper/2606.06205/) (2026)
- [An RKHS Framework for Fixed Effects in Permanental Process Models](https://mlsanity.com/paper/2608.17908/) (2026)

## Abstract (arXiv)

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.