Multivariate Poisson intensity estimation via low-rank tensor decomposition

2025-04-22

AI Summary

This paper addresses the nonparametric estimation of multivariate intensity functions for spatial point processes. To tackle the bias–variance trade-off arising from infinite-rank functional tensors, we propose the first modeling framework that integrates matrix and tensor methods in function spaces, featuring a computationally tractable low-rank functional tensor approximation and an efficient estimation algorithm based on matrix regularization. Our key contributions are threefold: (1) the first incorporation of low-rank tensor structure into spatial point process intensity modeling; (2) a theoretical characterization of infinite-rank functional tensors and a practical low-rank decomposition paradigm; and (3) preservation of statistically optimal convergence rates while substantially reducing computational complexity. Numerical experiments demonstrate consistent superiority over state-of-the-art estimators across multiple benchmark tasks.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In this work, we introduce new matrix- and tensor-based methodologies for estimating multivariate intensity functions of spatial point processes. By modeling intensity functions as infinite-rank tensors within function spaces, we develop new algorithms to reveal optimal bias-variance trade-off for infinite-rank tensor estimation. Our methods dramatically enhance estimation accuracy while simultaneously reducing computational complexity. To our knowledge, this work marks the first application of matrix and tensor techinques to spatial point processes. Extensive numerical experiments further demonstrate that our techniques consistently outperform current state-of-the-art methods.

Problem

• Estimating multivariate intensity functions of spatial point processes
• Developing algorithms for optimal bias-variance trade-off in tensor estimation
• Improving estimation accuracy while reducing computational complexity

Innovation

• Low-rank tensor decomposition for intensity estimation
• Optimal bias-variance trade-off algorithms
• Matrix and tensor techniques in spatial processes