  Higher-order multiway data is ubiquitous in machine learning and statistics
and often exhibits community-like structures, where each component (node) along
each different mode has a community membership associated with it. In this
paper we propose the tensor mixed-membership blockmodel, a generalization of
the tensor blockmodel positing that memberships need not be discrete, but
instead are convex combinations of latent communities. We establish the
identifiability of our model and propose a computationally efficient estimation
procedure based on the higher-order orthogonal iteration algorithm (HOOI) for
tensor SVD composed with a simplex corner-finding algorithm. We then
demonstrate the consistency of our estimation procedure by providing a per-node
error bound, which showcases the effect of higher-order structures on
estimation accuracy. To prove our consistency result, we develop the
$\ell_{2,\infty}$ tensor perturbation bound for HOOI under independent,
heteroskedastic, subgaussian noise that may be of independent interest. Our
analysis uses a novel leave-one-out construction for the iterates, and our
bounds depend only on spectral properties of the underlying low-rank tensor
under nearly optimal signal-to-noise ratio conditions such that tensor SVD is
computationally feasible. Finally, we apply our methodology to real and
simulated data, demonstrating some effects not identifiable from the model with
discrete community memberships.
