Permutation-preserving Functions and Neural Vecchia Covariance Kernels

2026-05-06

AI Summary

Modeling nonstationary Gaussian processes often entails a fundamental trade-off between expressiveness and computational scalability. This work proposes a novel approach that leverages deep neural networks to directly learn Kriging coefficients and conditional standard deviations under the Vecchia approximation, thereby constructing a flexible and scalable covariance kernel. The key innovation lies in a permutation-symmetric functional representation together with an equivariant neural architecture, which substantially improves training stability and data efficiency. By preserving linear computational complexity, the method effectively enhances the model’s capacity to capture complex nonstationary structures in spatial or temporal data.

Abstract

We introduce a novel framework for constructing scalable and flexible covariance kernels for Gaussian processes (GPs) by directly learning the covariance structure under a regression-type parameterization induced by Vecchia approximations, using deep neural architectures. Specifically, we model kriging coefficients and conditional standard deviations, deterministic quantities that uniquely characterize the covariance, providing stable and informative learning targets. Exploiting the permutation-equivariant structure of conditioning sets in the Vecchia factorization, we derive a universal representation for permutation-preserving functions and design neural architectures that respect this symmetry, leading to improved training stability and data efficiency. The proposed approach enables expressive, non-stationary kernel learning while maintaining computational scalability, thereby bridging classical GP methodology with modern deep learning.

Problem

• Gaussian processes
• covariance kernels
• non-stationary
• scalability
• Vecchia approximation

Innovation

• Permutation-equivariant
• Vecchia approximation
• Gaussian processes
• Neural covariance kernels
• Non-stationary kernel learning