Beyond Local Independence: High-Dimensional Latent Class Graphical Models with Shared Block Structure

Abstract

Latent class models are central tools for multivariate categorical data from heterogeneous populations, but their standard local-independence assumption is often unrealistic in modern high-dimensional applications. We propose a high-dimensional latent class graphical model for ordinal responses with block-structured local dependence. The model retains the interpretability and parsimony of classical latent class analysis by imposing a shared block partition of variables, while allowing class-specific graphical dependence within each block. We develop a scalable three-step estimator that first recovers latent classes by spectral clustering of a flattened response matrix, then estimates class-specific latent covariance matrices and aggregates them to recover the shared block partition, and finally estimates sparse within-block precision matrices. We establish finite-sample error bounds for clustering, covariance estimation, block recovery, and precision-matrix estimation, yielding end-to-end consistency of all model components under high-dimensional scaling. Simulations demonstrate accurate recovery of latent classes, the shared block partition, and class-specific dependence graphs with scalable computation. Applications to American National Election Studies survey data and HapMap3 genotype data show that the method uncovers interpretable local dependence structures while accounting for latent heterogeneity.