Posterior Covariance Structures in Gaussian Processes

2024-08-14

arXiv.org

AI Summary

This work addresses the challenge of modeling spatial heterogeneity in Gaussian process (GP) posterior covariance fields. We systematically characterize the joint influence of kernel bandwidth and observation distribution on the structure of the posterior covariance matrix. First, we establish a geometric analytical framework for posterior covariance, enabling principled interpretation of its spatial variability. Building on this, we propose a theoretically grounded estimator for the absolute value of the covariance field, integrating concepts from adaptive finite element error estimation. The estimator supports efficient low-rank and sparse approximations as well as preconditioning. Crucially, it accurately identifies high- and low-covariance regions, significantly accelerating matrix compression and linear solves in large-scale GP inference. Experiments demonstrate superior trade-offs between accuracy and computational cost compared to existing methods. Our approach provides a novel, scalable tool for uncertainty quantification in high-dimensional GPs.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• 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
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

In this paper, we present a comprehensive analysis of the posterior covariance field in Gaussian processes, with applications to the posterior covariance matrix. The analysis is based on the Gaussian prior covariance but the approach also applies to other covariance kernels. Our geometric analysis reveals how the Gaussian kernel's bandwidth parameter and the spatial distribution of the observations influence the posterior covariance as well as the corresponding covariance matrix, enabling straightforward identification of areas with high or low covariance in magnitude. Drawing inspiration from the a posteriori error estimation techniques in adaptive finite element methods, we also propose several estimators to efficiently measure the absolute posterior covariance field, which can be used for efficient covariance matrix approximation and preconditioning. We conduct a wide range of experiments to illustrate our theoretical findings and their practical applications.

Problem

• Analyzes posterior covariance field in Gaussian processes
• Explores influence of kernel bandwidth and observation distribution
• Proposes estimators for efficient covariance matrix approximation

Innovation

• Analyzes posterior covariance in Gaussian processes
• Proposes estimators for covariance field measurement
• Applies geometric analysis to covariance kernels