  Many applications involve estimation of a signal matrix from a noisy data
matrix. In such cases, it has been observed that estimators that shrink or
truncate the singular values of the data matrix perform well when the signal
matrix has approximately low rank. In this article, we generalize this approach
to the estimation of a tensor of parameters from noisy tensor data. We develop
new classes of estimators that shrink or threshold the mode-specific singular
values from the higher-order singular value decomposition. These classes of
estimators are indexed by tuning parameters, which we adaptively choose from
the data by minimizing Stein's unbiased risk estimate. In particular, this
procedure provides a way to estimate the multilinear rank of the underlying
signal tensor. Using simulation studies under a variety of conditions, we show
that our estimators perform well when the mean tensor has approximately low
multilinear rank, and perform competitively when the signal tensor does not
have approximately low multilinear rank. We illustrate the use of these methods
in an application to multivariate relational data.

Methodology