发表机构
Institute of Mathematical Statistics and Actuarial Science; University of Bern（数理统计与精算科学研究所; 伯尔尼大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出三重可扩展等变高斯过程，通过等变稀疏变分与无矩阵全GP实现，高效处理高维对称数据，并在分子性质预测中实现低计算成本的不确定性量化。
AI 中文摘要
高斯过程（GPs）在编码先验知识（包括等变性）的同时提供原则性的概率预测。然而，它们在大型科学问题中的应用受到计算成本的限制。等变神经网络很常见，但通常缺乏GP提供的不确定性量化，而这在分子研究等应用中很有价值。高维输入和大对称群进一步要求可扩展性。我们建立了关于GP等变性与条件化相互作用的结果，并利用这些结果通过合适的均值函数和协方差核获得等变稀疏GP。我们用一类灵活的免积分等变核实例化该框架，产生可扩展且数据高效的GP推断。特别是，我们引入了三重可扩展等变高斯过程。我们将等变稀疏变分高斯过程（SVGP）用于$\text{SO}(2)$等变向量场和分子性质预测。除了SVGP，我们还开发了一种无矩阵等变全GP实现，该实现将精确的Kronecker约简与预处理共轭梯度求解相结合，能够快速可扩展地评估全联合预测密度。我们进一步比较了在等变稀疏GP模型中选择诱导点的不同方法。我们的测试案例包括合成$\text{SO}(2)$等变场以及基于量子化学模拟的N-甲基甲酰胺电偶极矩预测，以经典GP推断计算成本的一小部分实现了准确、不确定性感知的预测。
英文摘要
Gaussian processes (GPs) provide principled probabilistic predictions while encoding prior knowledge, including equivariances. Yet, their use in large-scale scientific problems is limited by computational cost. Equivariant neural networks are common but typically lack the uncertainty quantification offered by GPs, which is valuable in applications such as molecular research. High-dimensional inputs and large symmetry groups further demand scalability. We establish results pertaining to the interplay of GP equivariance and conditioning and leverage them to obtain equivariant sparse GPs through suitable mean functions and covariance kernels. We instantiate this framework with a flexible class of integration-free equivariant kernels, yielding scalable and data-efficient GP inference. In particular, we introduce triply scalable equivariant Gaussian processes. We employ equivariant sparse variational Gaussian processes for $\mathrm{SO}(2)$-equivariant vector fields and molecular property prediction. Alongside the SVGP, we develop a matrix-free equivariant full-GP implementation that combines an exact Kronecker reduction with preconditioned conjugate-gradient solves, enabling fast and scalable evaluation of the full joint predictive density. We further compare different approaches for selecting inducing points in the equivariant sparse GP models. Our test cases include synthetic $\mathrm{SO}(2)$-equivariant fields as well as the prediction of electric dipole moments of N-methylformamide based on quantum chemistry simulations, achieving accurate, uncertainty-aware predictions at a fraction of the computational cost of classical GP inference.
