Abstract

Mixture models are effective tools for modeling data generated from multiple unobserved sources, and have been popular in the social and behavioral sciences as well as modern machine learning. This talk explores the estimation of two complex mixture models that both involve a latent graphical structure. In the first part, I consider Gaussian mixture models where labels exhibit network dependence, such as an Ising model. I illustrate the surprising rate-optimality of a naive estimator based on a misspecified likelihood, and propose a refined estimator that utilizes the dependence structure for superior performance. The second part considers another mixture model with a deep latent structure formulated as a directed graphical model. I discuss the identifiability of this model, which is a critical challenge in unsupervised learning. I also propose a computationally tractable method and provide illustrations in image representation learning and topic modeling. Finally, I conclude with broader implications of my work and other research directions, including applications in psychometrics and high-dimensional theoretical analysis of (empirical) Bayes procedures.