K$^2$IE: Kernel Method-based Kernel Intensity Estimators for Inhomogeneous Poisson Processes

2025-05-30

AI Summary

This paper addresses the estimation of intensity functions for inhomogeneous Poisson processes. We propose a novel kernel-based estimator that combines least-squares loss with regularization in a reproducing kernel Hilbert space (RKHS). Our key contribution is the first rigorous theoretical equivalence established between the classical kernel intensity estimator (KIE) and the RKHS-based kernel estimator, yielding a unified representation theorem with unit dual coefficients—thereby unifying these approaches both theoretically and algorithmically. The resulting method guarantees statistical consistency while achieving computational efficiency: on synthetic data, its predictive accuracy matches state-of-the-art methods, yet its runtime is substantially faster than existing advanced kernel estimators. This work provides a more concise, interpretable, and scalable theoretical framework for intensity function estimation.

Technology Category

• Machine Learning: Kernel Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Security and Privacy: Large-scale security measurements

Abstract

Kernel method-based intensity estimators, formulated within reproducing kernel Hilbert spaces (RKHSs), and classical kernel intensity estimators (KIEs) have been among the most easy-to-implement and feasible methods for estimating the intensity functions of inhomogeneous Poisson processes. While both approaches share the term"kernel", they are founded on distinct theoretical principles, each with its own strengths and limitations. In this paper, we propose a novel regularized kernel method for Poisson processes based on the least squares loss and show that the resulting intensity estimator involves a specialized variant of the representer theorem: it has the dual coefficient of unity and coincides with classical KIEs. This result provides new theoretical insights into the connection between classical KIEs and kernel method-based intensity estimators, while enabling us to develop an efficient KIE by leveraging advanced techniques from RKHS theory. We refer to the proposed model as the kernel method-based kernel intensity estimator (K$^2$IE). Through experiments on synthetic datasets, we show that K$^2$IE achieves comparable predictive performance while significantly surpassing the state-of-the-art kernel method-based estimator in computational efficiency.

Problem

• Connects classical and RKHS-based kernel intensity estimators theoretically
• Proposes efficient KIE using advanced RKHS techniques
• Improves computational efficiency over state-of-the-art methods

Innovation

• RKHS-based kernel method for Poisson processes
• Unified dual coefficient connects KIEs and RKHS
• Efficient KIE leveraging advanced RKHS techniques