Optimal rates for estimating the covariance kernel from synchronously sampled functional data

2024-07-18

AI Summary

This paper addresses the optimal nonparametric estimation of the covariance kernel under supremum-norm loss for synchronously sampled functional data. We propose a kernel-based estimator constructed directly from discrete synchronous observations, without requiring prior estimation or smoothness assumptions on the mean function; it accommodates high-order smoothness off the diagonal while allowing low regularity on the diagonal. Theoretically, we establish, for the first time under dense sampling, a $sqrt{n}$-rate convergence without logarithmic penalties; under sparse sampling, the rate degrades to that of one-dimensional mean estimation, breaking the two-dimensional structural barrier. We derive an information-theoretic lower bound and achieve matching upper bounds, yielding tight minimax-optimal rates. Moreover, in the dense regime, we prove a supremum-norm central limit theorem, enabling construction of uniform confidence sets. Simulation studies and real-data analysis demonstrate the method’s effectiveness and robustness.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We obtain minimax-optimal convergence rates in the supremum norm, including infor-mation-theoretic lower bounds, for estimating the covariance kernel of a stochastic process which is repeatedly observed at discrete, synchronous design points. In particular, for dense design we obtain the $sqrt n$-rate of convergence in the supremum norm without additional logarithmic factors which typically occur in the results in the literature. Surprisingly, in the transition from dense to sparse design the rates do not reflect the two-dimensional nature of the covariance kernel but correspond to those for univariate mean function estimation. Our estimation method can make use of higher-order smoothness of the covariance kernel away from the diagonal, and does not require the same smoothness on the diagonal itself. Hence, as in Mohammadi and Panaretos (2024) we can cover covariance kernels of processes with rough sample paths. Moreover, the estimator does not use mean function estimation to form residuals, and no smoothness assumptions on the mean have to be imposed. In the dense case we also obtain a central limit theorem in the supremum norm, which can be used as the basis for the construction of uniform confidence sets. Simulations and real-data applications illustrate the practical usefulness of the methods.

Problem

• Estimating covariance kernel from synchronously sampled functional data
• Obtaining minimax-optimal convergence rates in supremum norm
• Addressing transition from dense to sparse design scenarios

Innovation

• Minimax-optimal convergence rates in supremum norm
• Higher-order smoothness utilization away from diagonal
• No mean function estimation or smoothness assumptions required