Vecchia Gaussian Processes: on probabilistic and statistical properties
时间：2026-04-03
阅读量：630次
- 演讲人： 朱泆辰（香港大学，助理教授）
- 时间：2026年4月8日14:00
- 地点：浙江大学紫金港校区行政楼1417报告厅
- 主办单位：浙江大学数据科学研究中心
Abstract: Gaussian Processes (GPs) are widely used to model dependencies in spatial statistics and machine learning. However, exact inference is computationally intractable for GP regression, with a time complexity of $O(n^3)$. The Vecchia approximation scales up computation by introducing sparsity into the spatial dependency structure, represented by a directed acyclic graph (DAG). Despite its practical popularity, this approach lacks rigorous theoretical foundations, and the choice of DAG structure remains an open problem.
In this paper, we systematically study the Vecchia approximation of the popular, isotropic Mat\'{e}rn GP as standalone stochastic process and uncover key probabilistic and statistical properties. We propose selecting parent sets as norming sets with fixed cardinality in the Vecchia approximation. On the probabilistic side, we show that the conditional distributions of Matérn GPs, as well as their Vecchia approximations, can be characterized by polynomial interpolations. This enables us to establish several results on small ball probabilities and the Reproducing Kernel Hilbert Spaces (RKHSs) of Vecchia GPs. Building on these probabilistic results, we prove that in the nonparametric regression model, the corresponding posterior contracts around the truth at the optimal minimax rate, both under oracle rescaling and hierarchical tuning of the prior. We illustrate the theoretical findings through numerical experiments on synthetic datasets. Our core algorithms are implemented in C$++$ with an R interface.
Short Bio:
Bio: Dr. Yichen Zhu is currently a tenure-track Assistant Professor at the School of Computing and Data Science, the University of Hong Kong. Prior to that, he was a postdoctoral researcher at Bocconi University, advised by Prof. Botond Szabo. He obtained Ph.D. in Statistical Science at Duke University, advised by Prof. David B. Dunson. He obtained B.S. in mathematics at Peking University, advised by Prof. Wei Lin. His research interests span the broad areas of Bayesian Statistics, Deep Learning, Nonparametric Statistics, Statistical Computing and Theoretical Statistics. His research papers previously appear in the Annals of Statistics, the Journal of American Statistical Association, Biometrika, and ICML.