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Higher-order multiway data is ubiquitous in machine learning and statisticsand often exhibits community-like structures, where each component (node) alongeach different mode has a community membership associated with it. In thispaper we propose the tensor mixed-membership blockmodel, a generalization ofthe tensor blockmodel positing that memberships need not be discrete, butinstead are convex combinations of latent communities. We establish theidentifiability of our model and propose a computationally efficient estimationprocedure based on the higher-order orthogonal iteration algorithm (HOOI) fortensor SVD composed with a simplex corner-finding algorithm. We thendemonstrate the consistency of our estimation procedure by providing a per-nodeerror bound, which showcases the effect of higher-order structures onestimation accuracy. To prove our consistency result, we develop the$\ell_{2,\infty}$ tensor perturbation bound for HOOI under independent,possibly heteroskedastic, subgaussian noise that may be of independentinterest. Our analysis uses a novel leave-one-out construction for theiterates, and our bounds depend only on spectral properties of the underlyinglow-rank tensor under nearly optimal signal-to-noise ratio conditions such thattensor SVD is computationally feasible. Whereas other leave-one-out analysestypically focus on sequences constructed by analyzing the output of a givenalgorithm with a small part of the noise removed, our leave-one-out analysisconstructions use both the previous iterates and the additional tensorstructure to eliminate a potential additional source of error. Finally, weapply our methodology to real and simulated data, including applications to twoflight datasets and a trade network dataset, demonstrating some effects notidentifiable from the model with discrete community memberships.

Estimating Higher-Order Mixed Memberships via the $\ell_{2,\infty}$ Tensor Perturbation Bound