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On the Local Structure and Approximation Stability of Block Isotropic Gaussian Fields

Munki Jeong, Alexander Strang

TL;DR

The authors develop a rigorous framework for local quadratic approximations of block isotropic Gaussian fields, focusing on skew-symmetric constructions where traditional stationarity is incompatible with skew-symmetry. They show that the local geometry at matched inputs decomposes into a Gaussian gradient and Gaussian Hessian blocks (GOE and GSSE components), and that the second-order Taylor approximation remains Gaussian with an explicit covariance structure. By analyzing the error between the original field and its quadratic surrogate, they derive pointwise, regional (ellipsoidal), and distributional uniform bounds, including detailed asymptotics for small and large neighborhoods and dimension-dependent behavior. The results provide practical guidance for applying local quadratic approximations in high-dimensional settings, including how kernel smoothness and input distribution influence error magnitudes and how region sizes must be adapted with dimension to maintain accuracy. Overall, the work offers a tractable, distributional lens for understanding local approximations of skew-symmetric Gaussian fields and their errors in both theory and potential applications.

Abstract

Skew-symmetric functions are a class of functions defined on a product space $M \times M$ that are antisymmetric with respect to the order of their inputs. In [13], the authors proved that non-deterministic skew-symmetric Gaussian fields cannot be stationary or isotropic and proposed an alternative notion: stationarity (isotropy) in each component space. Our work focuses on local quadratic approximations of the associated Gaussian fields. Local quadratic approximations to random fields are random polynomials parametrized by a jointly sampled gradient vector and Hessian matrix. We characterize the distribution of the corresponding random vectors and random matrices. Then, we study the error in the quadratic approximation, which is also a Gaussian field. We investigate the error induced by the quadratic approximation in three senses: the pointwise error, the maximal error over an ellipsoidal region, and the worst-case error for multivariate Gaussian inputs at a given confidence level. Next, we explore the limiting behavior of the worst-case error as the distance between an expansion point and evaluation points approaches zero and infinity. Finally, we study how, as the input dimension increases, the variance of multivariate Gaussian distributions must be restricted to keep the worst-case error bound constant.

On the Local Structure and Approximation Stability of Block Isotropic Gaussian Fields

TL;DR

The authors develop a rigorous framework for local quadratic approximations of block isotropic Gaussian fields, focusing on skew-symmetric constructions where traditional stationarity is incompatible with skew-symmetry. They show that the local geometry at matched inputs decomposes into a Gaussian gradient and Gaussian Hessian blocks (GOE and GSSE components), and that the second-order Taylor approximation remains Gaussian with an explicit covariance structure. By analyzing the error between the original field and its quadratic surrogate, they derive pointwise, regional (ellipsoidal), and distributional uniform bounds, including detailed asymptotics for small and large neighborhoods and dimension-dependent behavior. The results provide practical guidance for applying local quadratic approximations in high-dimensional settings, including how kernel smoothness and input distribution influence error magnitudes and how region sizes must be adapted with dimension to maintain accuracy. Overall, the work offers a tractable, distributional lens for understanding local approximations of skew-symmetric Gaussian fields and their errors in both theory and potential applications.

Abstract

Skew-symmetric functions are a class of functions defined on a product space that are antisymmetric with respect to the order of their inputs. In [13], the authors proved that non-deterministic skew-symmetric Gaussian fields cannot be stationary or isotropic and proposed an alternative notion: stationarity (isotropy) in each component space. Our work focuses on local quadratic approximations of the associated Gaussian fields. Local quadratic approximations to random fields are random polynomials parametrized by a jointly sampled gradient vector and Hessian matrix. We characterize the distribution of the corresponding random vectors and random matrices. Then, we study the error in the quadratic approximation, which is also a Gaussian field. We investigate the error induced by the quadratic approximation in three senses: the pointwise error, the maximal error over an ellipsoidal region, and the worst-case error for multivariate Gaussian inputs at a given confidence level. Next, we explore the limiting behavior of the worst-case error as the distance between an expansion point and evaluation points approaches zero and infinity. Finally, we study how, as the input dimension increases, the variance of multivariate Gaussian distributions must be restricted to keep the worst-case error bound constant.

Paper Structure

This paper contains 29 sections, 41 theorems, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on Gaussian Random Fields
  3. Gaussian Random Fields
  4. Differentiability of Gaussian Random Fields
  5. Preliminaries on Random Vectors and Random Matrices
  6. Block Isotropic Gaussian Fields
  7. Skew-Symmetric Gaussian Fields
  8. Block Isotropic Gaussian Fields
  9. Local Quadratic Approximation of Block Isotropic Gaussian Processes
  10. Second-Order Taylor Approximation of Block Isotropic Gaussian Fields at Matched Inputs
  11. Distribution of the Local Geometrical Structures
  12. Error induced by Quadratic Approximation
  13. Error Fields of Block Isotropic Gaussian Fields
  14. Pointwise Error Bound
  15. Maximal Error over Ellipsoidal Region
  16. ...and 14 more sections

Key Result

Proposition 1

Let $R: \mathbb{R}^{2D} \to \mathbb{R}^{2D}$ be an arbitrary orthogonal transformation (e.g., rotation or reflection). If $\{u(x)\}_{x \in \mathbb{R}^{2D}}$ is an isotropic Gaussian field, then it is distributionally invariant under $Q$, i.e., Conversely, for a stationary Gaussian field, distributional invariance under all orthogonal transformations implies isotropy.

Figures (1)

  • Figure 1: Visual representation of the extraction of the sub-gradient $g$ and Hessian blocks.

Theorems & Definitions (102)

  • Definition 1: Real-Valued Random Field
  • Definition 2: Gaussian Random Field
  • Definition 3: Equality in Distribution
  • Definition 4: Mean and Covariance Function
  • Definition 5: Stationary and Isotropic Gaussian Field
  • Proposition 1: Isotropic Gaussian Field is Orthogonally Invariant
  • proof
  • Definition 7: $L^2$ Continuity
  • Definition 8: Directional Derivative
  • Definition 9: Mean-Square Differentiability
  • ...and 92 more