arXiv：cs.LG· Ameer Qaqish, Didong Li·· 4 小时前AI 评分39
RBF 核高斯过程最大似然估计的精确渐近理论
Sharp Asymptotic Theory of Maximum Likelihood Estimation for Gaussian Processes with an RBF Kernel
AI 导读
针对 RBF 核高斯过程，研究者首次给出固定域渐近下空间方差、长度尺度和 nugget 方差三者联合 MLE 的完整渐近刻画，证明了相合性、推导出三个参数的收敛速率并证明联合渐近正态性。这些收敛速率被证明是极小极大最优的。
正文
Abstract:Gaussian processes (GPs) are widely used across machine learning, spatial statistics, time-series analysis, optimization, Bayesian statistics, and scientific applications. A central component of a GP model is its kernel, which is typically specified through a parametric family. Among the most widely used choices is the radial basis function (RBF), also known as the squared exponential or Gaussian kernel, owing to its simple form, smoothness, and flexibility. In practice, the kernel parameters are routinely estimated by the maximum likelihood estimators (MLEs), as implemented by standard GP software. Despite this widespread use, the asymptotic behavior of the MLEs remains poorly understood under fixed-domain asymptotics, even for the RBF kernel. The main difficulty arises from the increasingly strong dependence among densely sampled observations and the nonlinear dependence of the covariance matrix on the kernel parameters. In this paper, we address this gap by providing, to the best of our knowledge, the first complete asymptotic characterization of the joint MLE of the spatial variance, lengthscale, and nugget variance under fixed-domain asymptotics. We establish consistency, derive convergence rates for all three parameters, prove joint asymptotic normality, and
show that these rates are minimax optimal.
| Subjects: | Statistics Theory (math.ST); Machine Learning (cs.LG); Probability (math.PR) |
| Cite as: | arXiv:2610.10080 [math.ST] |
| (or arXiv:2610.10080v1 [math.ST] for this version) | |
| https://doi.org/10.48550/arXiv.2610.10080 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Didong Li [view email]
[v1]
Wed, 7 Oct 2026 13:44:55 UTC (1,222 KB)
来源：arXiv：cs.LG · arxiv.org