发表机构
Division of Statistics and Data Science, Department of Mathematical Sciences, University of Cincinnati; Department of Mathematics, Purdue University（统计与数据科学系，数学科学系，辛辛那提大学; 数学系，普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对高斯过程建模中高维输入难题，本文引入基于分层贝叶斯模型的框架，集成降维与建模推理，通过哈密顿蒙特卡洛实现后验推理，并结合深度高斯过程扩展框架，提升了预测性能与不确定性量化。
AI 中文摘要
高斯过程（GP）建模在计算科学与工程中广泛应用。但因维度诅咒，将GP应用于高维输入仍具挑战。现有降维方法多为两阶段，分别进行降维和GP拟合。本文引入贝叶斯框架，将降维与GP建模及推理无缝集成。该方法基于带先验的分层贝叶斯模型，在Stiefel流形上强制投影矩阵正交归一，并通过哈密顿蒙特卡洛与测地线流进行后验推理。此外，通过结合带内置降维的深度高斯过程（DGP）扩展框架。数值研究表明，虽该贝叶斯方法计算成本更高，但提升了预测性能和不确定性量化，为现有方法提供了稳健替代。
英文摘要
Gaussian process (GP) modeling is widely used in computational science and engineering. However, fitting a GP to high-dimensional inputs remains challenging due to the curse of dimensionality. While various methods have been proposed to reduce input dimensionality, they typically follow a two-stage approach, performing dimension reduction and GP fitting separately. We introduce a Bayesian framework that seamlessly integrates dimensionality reduction with GP modeling and inference. Our approach, built on a hierarchical Bayesian model with priors on the Stiefel manifold, enforces orthonormality on the projection matrix and enables posterior inference via Hamiltonian Monte Carlo with geodesic flow. Additionally, we extend this framework by incorporating Deep Gaussian Processes (DGP) with built-in dimension reduction, providing a more flexible and powerful tool for complex datasets. Through extensive numerical studies, we demonstrate that while the proposed Bayesian method incurs higher computational costs, it improves predictive performance and uncertainty quantification, providing a principled and robust alternative to existing methods.
