  Many modern datasets, from areas such as neuroimaging and geostatistics, come
in the form of a random sample of tensor-valued data which can be understood as
noisy observations of a smooth multidimensional random function. Most of the
traditional techniques from functional data analysis are plagued by the curse
of dimensionality and quickly become intractable as the dimension of the domain
increases. In this paper, we propose a framework for learning continuous
representations from a sample of multidimensional functional data that is
immune to several manifestations of the curse. These representations are
constructed using a set of separable basis functions that are defined to be
optimally adapted to the data. We show that the resulting estimation problem
can be solved efficiently by the tensor decomposition of a carefully defined
reduction transformation of the observed data. Roughness-based regularization
is incorporated using a class of differential operator-based penalties.
Relevant theoretical properties are also established. The advantages of our
method over competing methods are demonstrated in a simulation study. We
conclude with a real data application in neuroimaging.
