  Interest in higher-order tensors has recently surged in data-intensive
fields, with a wide range of applications including image processing, blind
source separation, community detection, and feature extraction. A common
paradigm in tensor-related algorithms advocates unfolding (or flattening) the
tensor into a matrix and applying classical methods developed for matrices.
Despite the popularity of such techniques, how the functional properties of a
tensor changes upon unfolding is currently not well understood. In contrast to
the body of existing work which has focused almost exclusively on
matricizations, we here consider all possible unfoldings of an order-$k$
tensor, which are in one-to-one correspondence with the set of partitions of
$\{1,\ldots,k\}$. We derive general inequalities between the $l^p$-norms of
arbitrary unfoldings defined on the partition lattice. In particular, we
demonstrate how the spectral norm ($p=2$) of a tensor is bounded by that of its
unfoldings, and obtain an improved upper bound on the ratio of the Frobenius
norm to the spectral norm of an arbitrary tensor. For specially-structured
tensors satisfying a generalized definition of orthogonal decomposability, we
prove that the spectral norm remains invariant under specific subsets of
unfolding operations.
