Finding mixed memberships in categorical data
Latent class analysis is a fundamental problem in categorical data analysis, which often encounters overlapping latent classes that further challenge the problem. This paper addresses the problem of finding latent mixed memberships of subjects in categorical data with polytomous responses under the Grade of Membership (GoM) model, which allows each subject to be associated with a membership score in each latent class. We propose two efficient spectral algorithms for estimating latent mixed memberships and other GoM parameters. Our algorithms are developed by using the singular value decomposition of a regularized Laplacian matrix. We establish their convergence rates under a mild condition on data sparsity. We also provide a measure to evaluate the quality of estimated mixed memberships for real-world categorical data and determine the number of latent classes based on this measure. Finally, we demonstrate the performances of our methods in both computer-generated and real-world categorical data.