  Grade of Membership (GoM) models are popular individual-level mixture models
for multivariate categorical data. GoM allows each subject to have mixed
memberships in multiple extreme latent profiles. Therefore GoM models have a
richer modeling capacity than latent class models that restrict each subject to
belong to a single profile. The flexibility of GoM comes at the cost of more
challenging identifiability and estimation problems. In this work, we propose a
singular value decomposition (SVD) based spectral approach to GoM analysis with
multivariate binary responses. Our approach hinges on the observation that the
expectation of the data matrix has a low-rank decomposition under a GoM model.
For identifiability, we develop sufficient and almost necessary conditions for
a notion of expectation identifiability. For estimation, we extract only a few
leading singular vectors of the observed data matrix, and exploit the simplex
geometry of these vectors to estimate the mixed membership scores and other
parameters. We also establish the consistency of our estimator in the
double-asymptotic regime where both the number of subjects and the number of
items grow to infinity. Our spectral method has a huge computational advantage
over Bayesian or likelihood-based methods and is scalable to large-scale and
high-dimensional data. Extensive simulation studies demonstrate the superior
efficiency and accuracy of our method. We also illustrate our method by
applying it to a personality test dataset.
