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.
Rolling covariance estimates feed two objects that are routinely treated as market structure. The first is the dominant eigenspace, monitored through the projector movement $\widehat D_{K,t}=\|\widehat P_{K,t}-\widehat P_{K,t-1}\|_F$; the second comprises scalar spectral functionals such as the absorption ratio and the...
Local polynomial smoothing is commonly used in non-parametric regression, but local linear derivative estimation still has a bias of order $O(h^2)$. This paper proposes an iterative data sharpening method to reduce the bias of derivative estimates while retaining the simplicity of local linear fitting. The method is ba...
Together, the results clarify how the input geometry shapes the kernel features and fundamentally impacts its generalization properties and clarify how the input geometry shapes the kernel features and fundamentally impacts its generalization properties.
L. Rizzi, Arie Wortsman Zurich, Bruno Loureiro· 0 citations
A novel approach is proposed, referred to as pooling ridge estimation, which combines the advantages of pooling strategy and RKHS-based method by incorporating the unbiased estimation of operators based on discretely observed measurements from all subjects, to achieve minimax optimality in prediction risk in arbitrary...
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 fixe...
This work proposes a Gaussian refit for kernel ridge regression by Anderson's inequality, which requires no moment assumptions and is calibrated at any confidence level via order statistics, and extends empirically to nonlinear constrained estimators and real spatial data.
Yijin Ni, Xiaoming Huo· 2 citations
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