  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.
