We study nonparametric covariance function estimation for functional data observed with noise at discrete locations on a $d$-dimensional domain. Estimating the covariance function from discretely observed data is a challenging nonparametric problem, particularly in multidimensional settings, since the covariance function is defined on a product domain and thus suffers from the curse of dimensionality. This motivates the use of adaptive estimators, such as deep learning estimators. However, existing theoretical results are largely limited to estimators with explicit analytic representations, and the properties of general learning-based estimators remain poorly understood. We establish an oracle inequality for a broad class of learning-based estimators that applies to both sparse and dense observation regimes in a unified manner, and derive convergence rates for deep learning estimators over several classes of covariance functions. The resulting rates suggest that structural adaptation can mitigate the curse of dimensionality, similarly to classical nonparametric regression. We further compare the convergence rates of learning-based estimators with several existing procedures. For a one-dimensional smoothness class, deep learning estimators are suboptimal, whereas local linear smoothing estimators achieve a faster rate. For a structured function class, however, deep learning estimators attain the minimax rate up to polylogarithmic factors, whereas local linear smoothing estimators are suboptimal. These results reveal a distinctive adaptivity-variance trade-off in covariance function estimation.

翻译：我们研究了在 $d$ 维域上离散位置含噪声观测的函数型数据中，协方差函数的非参数估计问题。从离散观测数据估计协方差函数是一个具有挑战性的非参数问题，尤其是在多维场景下，因为协方差函数定义在乘积域上，从而受到维度诅咒的影响。这激发了自适应估计量（如深度学习估计量）的应用。然而，现有理论结果主要局限于具有显式解析表示的估计量，而基于学习的通用估计量的性质仍知之甚少。我们针对一类广泛的基于学习的估计量建立了风险上界不等式，该不等式以统一的方式适用于稀疏和密集观测机制，并推导了深度学习估计量在若干类协方差函数上的收敛速率。所得速率表明，结构自适应可缓解维度诅咒，类似于经典的非参数回归。我们进一步将基于学习的估计量的收敛速率与若干现有方法进行了比较。对于一维光滑类，深度学习估计量是次优的，而局部线性平滑估计量实现了更快的速率。然而，对于结构化函数类，深度学习估计量达到了极小化最优速率（至多相差多对数因子），而局部线性平滑估计量则是次优的。这些结果揭示了协方差函数估计中独特的适应性-方差权衡。