Low-Rank Covariance Function Estimation for Multidimensional Functional Data

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.

如今，多维函数数据广泛涌现于众多研究领域。协方差函数在这类日益普及的数据的分析中发挥着关键作用。本文在再生核希尔伯特空间（reproducing kernel Hilbert spaces, RKHS）框架下，提出一种全新的非参数协方差函数估计方法，可同时适配稀疏与稠密函数型数据。我们将（有限维）张量的多线性秩结构推广至函数空间，从而能够灵活建模协方差算子与边缘结构。所提框架可确保所得估计量自动满足半正定性，同时能够兼容各类谱正则化策略。其中，迹范数正则化尤其能够推动协方差算子与边缘结构向低秩形式收敛。尽管该方法缺乏闭式解，但在温和假设条件下，所提估计量可获得统一的理论结果，该结果适用于样本量与每个样本场的观测数之间的任意相对规模，且收敛速率揭示了从稀疏到稠密函数型数据的相变现象。基于全新的代表定理，本文针对迹范数正则化设计了交替方向乘子法（ADMM）求解算法。模拟实验与阿尔戈（Argo）项目数据集的实证分析，验证了所提估计量优异的数值表现。本文补充材料可在线获取。