AI 中文总结
本研究针对周期域拟均匀设计下的Matérn回归，建立有限样本理论，明确目标与核长度尺度对回归精度的影响，揭示分辨率条件与光滑性对恢复误差的作用规律。
AI 中文摘要
核回归的理论保证通常以光滑性来表述，但实际精度关键取决于设计分辨率与目标尺度、核长度尺度的对比关系。我们针对周期域上的Matérn回归（含无噪插值与带噪核岭回归），在拟均匀设计下建立有限样本理论。我们的极小极大结果表明，准确恢复需要足够密集的设计以分辨目标尺度，且该尺度下有足够信息克服噪声。我们证明，Matérn插值还要求设计分辨核长度尺度：若核长度尺度相对点间距过短，即使目标已被良好分辨，仍会存在额外误差。对于带噪Matérn回归，我们推导了三项固定岭风险表征，包含目标尺度偏差、核尺度偏差与方差，且对岭参数优化会产生四种不同贡献。当目标光滑性不超过核的两倍时，选择比目标更长的核长度尺度不会恶化先验风险，而选择过短则会导致风险上升。因此，这些分辨率条件决定了准确恢复何时成为可能，而光滑性决定了误差后续下降的速率。
英文摘要
Theoretical guarantees for kernel regression are typically formulated in terms of smoothness, but practical accuracy depends critically on how the design resolution compares with the target and kernel lengthscales. We develop a finite-sample theory for Matérn regression on periodic domains with quasi-uniform designs, covering noiseless interpolation and noisy kernel ridge regression. Our minimax result shows that accurate recovery requires a design dense enough to resolve the target lengthscale and sufficient information at that scale to overcome noise. We prove that Matérn interpolation additionally requires the design to resolve the kernel lengthscale: if the kernel is too short relative to the point spacing, an additional error remains even when the target is well resolved. For noisy Matérn regression, we derive a three-term fixed-ridge risk characterization consisting of target-scale bias, kernel-scale bias, and variance, and show that optimizing over the ridge parameter yields four distinct contributions. When the target is no more than twice as smooth as the kernel, choosing a kernel lengthscale longer than the target does not worsen the oracle risk, whereas choosing one too short can. Thus these resolution conditions determine when accurate recovery becomes possible, while smoothness determines how rapidly the error decreases thereafter.
