Finite-rank approximations are widely used to scale Gaussian process (GP) regression, but their posterior behavior can differ from that of the corresponding parent GP prior. We study a class of finite-rank GP priors built from locally supported basis expansions with dependent Gaussian coefficients. Our framework covers finite-element approximations based on the stochastic partial differential equation (SPDE) representation of Matérn GPs and regular-grid GP interpolation schemes. We show that, with a suitable prior on the resolution parameter $N$, these finite-rank expansions inherit the same posterior contraction rate as the corresponding parent GP prior under the same bandwidth specification used for that parent prior. Consequently, the interpolation construction under a squared-exponential parent GP attains the minimax-optimal rate up to logarithmic factors under a hierarchical prior on the bandwidth parameter and on $N$, while the SPDE construction attains the same rate under a bandwidth scaling depending on the sample size and the smoothness of the true function, together with a prior on $N$. We also develop a posterior sampler for the hierarchical interpolation model that jointly updates the resolution and bandwidth parameters, and we provide numerical studies that support the theory.

翻译：有限秩近似被广泛用于扩展高斯过程（GP）回归，但其后验行为可能与对应父GP先验不同。我们研究了一类基于具有依赖高斯系数的局部支撑基展开的有限秩GP先验。我们的框架涵盖了基于马特恩（Matérn）GP的随机偏微分方程（SPDE）表示的有限元近似和规则网格GP插值方案。我们证明，在分辨率参数$N$上采用合适的先验条件下，当使用与父先验相同的带宽规范时，这些有限秩展开继承了对应父GP先验的后验收缩率。因此，在平方指数父GP下，基于带宽参数和$N$的分层先验，插值构造达到了对数因子下的极小极大最优速率；而在SPDE构造下，基于依赖于样本量和真实函数光滑度的带宽缩放以及$N$上的先验，实现了相同速率。我们还为分层插值模型开发了联合更新分辨率和带宽参数的后验采样器，并提供了支持该理论的数值实验。