Minimax Optimal Estimation of Mean and Covariance Functions with Spectral Regularization

TL;DR

This paper introduces a spectral regularization framework for estimating mean and covariance functions in functional data analysis, achieving optimal convergence rates without requiring the target functions to be in the RKHS.

Contribution

It develops a broad spectral regularization approach under H"{o}lder conditions, extending previous methods by relaxing the assumption that functions belong to the RKHS.

Findings

Derived convergence rates for estimators.

Established minimax optimality with matching lower bounds.

Applicable to a wide class of smoothness assumptions.

Abstract

Estimation of the mean and covariance functions is a fundamental problem in functional data analysis, particularly for discretely observed functional data. In this work, we study a regularization-based framework for estimating the mean and the covariance functions within a reproducing kernel Hilbert space (RKHS) setting. Our approach utilizes a spectral regularization technique under H\"{o}lder-type source conditions, allowing for a broad class of regularization schemes and accommodating a wide range of smoothness assumptions on the target functions. Unlike previous works in the literature, the proposed work does not require the target functions to belong to the underlying RKHS. Convergence rates for the proposed estimators are derived, and optimality is established by obtaining matching minimax lower bounds.