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Posterior Covariance Structures in Gaussian Processes

Difeng Cai, Edmond Chow, Yuanzhe Xi

TL;DR

This work analyzes the posterior covariance structure $R_{S,\rho}(\cdot,\cdot)$ in Gaussian processes, focusing on how kernel bandwidth $\rho$ and the observation set $S$ shape uncertainty through the posterior covariance field and matrix. It develops a two-regime geometric theory (small $\rho$ vs large $\rho$), derives explicit bounds, and introduces computable a posteriori–like indicators to locate regions of high covariance without inverting $K_{SS}$. The authors demonstrate practical benefits via extensive numerical experiments, including covariance-based matrix approximations, low-rank–plus–sparse corrections, and preconditioning strategies, with extensions to non-Gaussian covariances and noisy data. The results offer scalable tools for uncertainty quantification and efficient GP computations, particularly when bandwidth is small or data are irregular, and lay groundwork for broader applications and future enhancements in covariance-aware numerical linear algebra.

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.

Posterior Covariance Structures in Gaussian Processes

TL;DR

This work analyzes the posterior covariance structure in Gaussian processes, focusing on how kernel bandwidth and the observation set shape uncertainty through the posterior covariance field and matrix. It develops a two-regime geometric theory (small vs large ), derives explicit bounds, and introduces computable a posteriori–like indicators to locate regions of high covariance without inverting . The authors demonstrate practical benefits via extensive numerical experiments, including covariance-based matrix approximations, low-rank–plus–sparse corrections, and preconditioning strategies, with extensions to non-Gaussian covariances and noisy data. The results offer scalable tools for uncertainty quantification and efficient GP computations, particularly when bandwidth is small or data are irregular, and lay groundwork for broader applications and future enhancements in covariance-aware numerical linear algebra.

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.
Paper Structure (23 sections, 7 theorems, 67 equations, 25 figures, 2 tables)

This paper contains 23 sections, 7 theorems, 67 equations, 25 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminary Observations
  4. Theoretical Analysis and Posterior Covariance Indicators
  5. Small Bandwidth Case
  6. Large Bandwidth Case
  7. The "boundary effect"
  8. Illustration of the bounds in different cases
  9. Noisy observation case: $\tau>0$
  10. Geometric Posterior Covariance Indicators
  11. Numerical Experiments
  12. Pattern of posterior covariance: uniform data
  13. Pattern of posterior covariance: non-uniform data
  14. On Figure \ref{['fig:nf-sigma005']}: $\rho=0.05$
  15. On Figure \ref{['fig:nf-sigma01']}: $\rho=0.1$
  16. ...and 8 more sections

Key Result

Theorem 1

\newlabelthm:vanish0 For any finite subset $S\subseteq\mathbb{R}^d$, define Then for any $x,y$,

Figures (25)

  • Figure 1: $|R_{S,\rho}(x,y)|$ over $[0,1]\times [0,1]$: different $\rho$ and different $S$ (5 blue dots).
  • Figure 1: $|R_{S,\rho}(x,y)|$ over $[0,1]\times [0,1]$ with $\rho=0.1$. Blue triangles enclose locations with $\lVert x-y\rVert/\rho\geq 1$.
  • Figure 1: $|R_{S,\rho}(x,y)|$ over $[0,1]\times [0,1]$ with uniform$S$: different $\rho$.
  • Figure 2: $|R_{S,\rho}(x,y)|$ over $[0,1]\times [0,1]$ with $\rho=0.1$.
  • Figure 2: Function $|R_{S,\rho}(x,y)|$ over $[0,1]\times [0,1]$ with non-uniform$S$: different $\rho$.
  • ...and 20 more figures

Theorems & Definitions (14)

  • Theorem 1
  • Proof 1
  • Lemma 2
  • Proof 2
  • Theorem 3
  • Proof 3
  • Proposition 4
  • Proof 4
  • Theorem 5
  • Proof 5
  • ...and 4 more