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 article, 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 semipositive 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. Supplementary materials for this article are available online.

Keywords:

• Functional data analysis
• Multilinear ranks
• Tensor product space
• Unified theory

Supplementary Materials

In the supplementary materials related to this article, we provide formal definitions related to Tucker decomposition for finite-dimensional tensors, proofs of our theoretical findings and additional simulation results.

Acknowledgments

The authors would like to thank Professor Yehua Li for insightful discussions, and the editor, an associate editor and two referees for their constructive comments and suggestions.

Notes

¹ The n-mode product (Definition S1) is extended to the case when q_n is infinite.

Additional information

Funding

Raymond K. W. Wong’s research is partially supported by the National Science Foundation under grants DMS-1806063, DMS-1711952 (subcontract), and CCF-1934904. Xiaoke Zhang’s research is partially supported by the National Science Foundation under grant DMS-1832046. Portions of this research were conducted with high performance research computing resources provided by Texas A&M University (https://hprc.tamu.edu).