Low-Rank Covariance Function Estimation for Multidimensional Functional Data

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.

25 pages, 4 figures

References in corpus (4)

• Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data
• A reproducing kernel Hilbert space approach to functional linear regression
• A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization
• Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework

Cited by in corpus (5)

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• CovNet: Covariance Networks for Functional Data on Multidimensional Domains
• Functional Linear Regression with Mixed Predictors
• Robust Functional Principal Component Analysis for Non-Gaussian Longitudinal Data
• Row-clustering of a Point Process-valued Matrix