用神经网络学习可变序的协方差核，提升高斯过程的灵活性与效率
Permutation-preserving Functions and Neural Vecchia Covariance Kernels
- 通过神经网络直接学习克里金系数和条件标准差，构建稳定的学习目标
- 利用条件集的置换等变结构，设计对称神经架构，提升训练稳定性和数据效率
- 适合需要非平稳核且追求计算高效的研究者，如时空建模
我们提出一种新框架，通过深度神经网络直接学习高斯过程的协方差结构，基于向量化近似诱导的回归型参数化。具体而言，我们建模克里金系数和条件标准差——完全表征协方差的确定性量，提供稳定且信息丰富的学习目标。利用向量化分解中条件集的置换等变特性，我们推导出置换保持函数的通用表示，并设计符合该对称性的神经架构，从而提升训练稳定性与数据效率。该方法在保持计算可扩展性的前提下实现表达性强的非平稳核学习，成功将经典高斯过程方法与现代深度学习相融合。
原文摘要 · Abstract (English)
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.
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