AI 中文总结
研究标量对函数线性回归在有限噪声点评估下的极小极大估计与预测风险，推导两种采样方案下的速率，构建自适应估计器，揭示不同设计下的相变及丰富相图，数值模拟和实际数据示例验证结果。
AI 中文摘要
我们研究标量对函数的线性回归，此时每个协变量曲线仅通过有限多个有噪声的点评估来观测。目标是刻画极小极大估计和预测风险，作为轨迹数量\(n\)和轨迹内分辨率\(m\)的联合函数。在固定的三角特征基下，协方差特征值以速率\(\alpha\)衰减且斜率函数具有Sobolev光滑度\(s\)，我们在两种典型采样方案下推导了匹配的极小极大上界和下界。在独立随机设计下，极小极大预测速率为\(n^{-\frac{2\alpha + 2s}{2\alpha + 2s + 1}} + (nm)^{-\frac{2\alpha + 2s}{4\alpha + 2s + 1}}\)。在等距网格的共同设计下，极小极大预测速率变为\(n^{-\frac{2\alpha + 2s}{2\alpha + 2s + 1}} + (nm)^{-\frac{2\alpha + 2s}{4\alpha + 2s + 1}} + m^{-(2\alpha + 2s)} + m^{-4\alpha}\)。我们还构建了数据驱动的自适应估计器，无需事先了解特征值序列或光滑度指标就能达到这些速率。数值模拟和实际数据示例说明了理论结果。
英文摘要
We study scalar-on-function linear regression when each covariate curve is observed only through finitely many noisy point evaluations. Our goal is to characterize the minimax estimation and prediction risks as joint functions of the number of trajectories $n$ and the within-trajectory resolution $m$. Working in a fixed trigonometric eigenbasis, with covariance eigenvalues decaying at rate $α$ and slope function of Sobolev smoothness $s$, we derive matching minimax upper and lower bounds under two canonical sampling schemes. Under an independent random design, the minimax prediction rate is $n^{-\frac{2α+2s}{2α+2s+1}} + (nm)^{-\frac{2α+2s}{4α+2s+1}}$. The first term is the fully observed functional linear regression benchmark, while the second term captures the cost of noisy point evaluations after amplification by the inverse covariance operator. Under a common design on an equally spaced grid, the shared sampling geometry introduces additional obstructions, and the minimax prediction rate becomes $n^{-\frac{2α+2s}{2α+2s+1}} + (nm)^{-\frac{2α+2s}{4α+2s+1}} + m^{-(2α+2s)} + m^{-4α}$. Here the third term represents discretization error induced by the fixed grid, whereas the fourth reflects the cost of identifying unknown eigenvalues from observations on a common grid. We further construct data-driven adaptive estimators that screen the covariance scale and threshold blockwise prediction energy, attaining these rates without prior knowledge of the eigenvalue sequence or the smoothness indices. The results reveal a sharp phase transition that depends on the sampling resolution under independent design and a richer phase diagram under common design. Numerical simulations and a real data example illustrate the theoretical findings.
