  The latent class model is a widely used mixture model for multivariate
discrete data. Besides the existence of qualitatively heterogeneous latent
classes, real data often exhibit additional quantitative heterogeneity nested
within each latent class. The modern latent class analysis also faces extra
challenges, including the high-dimensionality, sparsity, and heteroskedastic
noise inherent in discrete data. Motivated by these phenomena, we introduce the
Degree-heterogeneous Latent Class Model and propose a spectral approach to
clustering and statistical inference in the challenging high-dimensional sparse
data regime. We propose an easy-to-implement HeteroClustering algorithm. It
uses heteroskedastic PCA with L2 normalization to remove degree effects and
perform clustering in the top singular subspace of the data matrix. We
establish an exponential error rate for HeteroClustering, leading to exact
clustering under minimal signal-to-noise conditions. We further investigate the
estimation and inference of the high-dimensional continuous item parameters in
the model, which are crucial to interpreting and finding useful markers for
latent classes. We provide comprehensive procedures for global testing and
multiple testing of these parameters with valid error controls. The superior
performance of our methods is demonstrated through extensive simulations and
applications to three diverse real-world datasets from political voting
records, genetic variations, and single-cell sequencing.
