用于泛函矩估计的核岭回归的精确泛化误差曲线
Exact Generalization Error Curves of Kernel Ridge Regression for Functional Moment Estimation
AI总结：
研究张量积核岭回归用于估计随机函数第r阶矩函数，给出L2误差精确展开，识别误差结构，表明KRR在源平滑度s≤2时达极小极大率，s>2时因饱和次优，还给出适用于功能观测相关结构的U统计量集中不等式。
AI中文摘要：
核岭回归是功能数据分析的标准方法，但其确切行为尚不太清楚。我们研究了张量积核岭回归，用于基于有噪声的离散观测估计随机函数的第r阶矩函数。该公式在单个框架中包括均值估计、协方差估计和高阶矩估计。我们的主要结果给出了每个可允许正则化参数处L2误差的精确1 + oP(1)展开。该展开由偏差和三个方差项组成，分别对应于独立样本路径的变化、每个样本点的潜在信号变化和测量误差的变化，识别了功能数据背后的精细误差结构。作为应用，我们表明KRR在源平滑度s≤2时达到极小极大率，但在s>2的稀疏情况下由于饱和而变得次优。一个技术要素是一组适用于功能观测的相关乘积结构的U统计量的集中不等式。
英文摘要：
Kernel ridge regression is a standard method for functional data analysis, but its exact behavior is less understood. We study tensor-product kernel ridge regression for estimating the $r$-th moment function of a random function based on noisy discrete observations. The formulation includes mean estimation, covariance estimation, and higher-order moment estimation in a single framework. Our main result gives a precise $1+o_{\mathbb{P}}(1)$ expansion for the $L^2$ error at each admissible regularization parameter. The expansion consists of bias and three variance terms corresponding respectively to variation across the independent sample paths, latent signal variation at each sample point, and variation from measurement errors, identifying the refined error structure underlying functional data. As applications, we show that KRR attains the minimax rate for source smoothness $s \leq 2$ but becomes suboptimal in the sparse regime for $s>2$ due to saturation. A technical ingredient is a set of concentration inequalities for $U$-statistics suited to the dependent product structure of functional observations.
