# Posterior Covariance Structures in Gaussian Processes

arXiv: https://arxiv.org/abs/2408.07379 · HTML: https://mlsanity.com/paper/2408.07379/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 43 references that are in the map.
- First posted on arXiv 14 Aug 2024.

## Builds on (most cited first, 3 of 3)

- [A general framework for Vecchia approximations of Gaussian processes](https://mlsanity.com/paper/1708.06302/) (2017)
- [Local Gaussian process approximation for large computer experiments](https://mlsanity.com/paper/1303.0383/) (2013)
- [Data-Driven Linear Complexity Low-Rank Approximation of General Kernel Matrices: A Geometric Approach](https://mlsanity.com/paper/2212.12674/) (2022)

## Built on by (most cited first, 2 of 2)

- [Preconditioned Additive Gaussian Processes with Fourier Acceleration](https://mlsanity.com/paper/2504.00480/) (2025)
- [Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations](https://mlsanity.com/paper/2506.23344/) (2025)

## Abstract (arXiv)

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.