Low-Rank Covariance Function Estimation for Multidimensional Functional Data

TL;DR

This paper introduces a nonparametric method for estimating covariance functions in multidimensional functional data using RKHS, capable of handling both sparse and dense data with theoretical guarantees and efficient algorithms.

Contribution

It extends multilinear rank structures to functions, ensuring positive semi-definiteness and incorporating spectral regularizations, including trace-norm for low-rank estimation.

Findings

Achieves unified convergence rates with phase-transition behavior.

Develops an ADMM algorithm for trace-norm regularization.

Demonstrates superior numerical performance in simulations and real data analysis.

Abstract

Multidimensional function data arise from many fields nowadays. The covariance function plays an important role in the analysis of such increasingly common data. In this paper, we propose a novel nonparametric covariance function estimation approach under the framework of reproducing kernel Hilbert spaces (RKHS) that can handle both sparse and dense functional data. We extend multilinear rank structures for (finite-dimensional) tensors to functions, which allow for flexible modeling of both covariance operators and marginal structures. The proposed framework can guarantee that the resulting estimator is automatically semi-positive definite, and can incorporate various spectral regularizations. The trace-norm regularization in particular can promote low ranks for both covariance operator and marginal structures. Despite the lack of a closed form, under mild assumptions, the proposed estimator can achieve unified theoretical results that hold for any relative magnitudes between the sample size and the number of observations per sample field, and the rate of convergence reveals the "phase-transition" phenomenon from sparse to dense functional data. Based on a new representer theorem, an ADMM algorithm is developed for the trace-norm regularization. The appealing numerical performance of the proposed estimator is demonstrated by a simulation study and the analysis of a dataset from the Argo project.