A PRIISM Seminar by Columbia's Yuqi Gu
Join PRIISM and Yuqi Gu to learn about an alternative to Latent Class Analysis, the Grade of Membership model and its estimation with multivariate categorical data.
Abstract
Mixed membership models are popular individual-level mixture models widely used in various fields including network analysis, topic modeling, and multivariate categorical data analysis. This work focuses on mixed membership models for multivariate categorical data, which are also called Grade of Membership (GoM) models. GoM models drastically increase the modeling flexibility of latent class models by allowing each individual to partially belong to multiple extreme latent profiles. However, such flexibility also comes with challenging identifiability and estimation issues, especially for high-dimensional polytomous (categorical with more than two categories) data. Such data take the form of a three-way tensor, with N subjects responding to J items each with C categories. Existing estimation methods based on maximum likelihood or Bayesian MCMC inference are not computationally efficient and lack high-dimensional theoretical guarantees. We propose an SVD-based spectral method for high-dimensional polytomous Models with potential local dependence. We innovatively flatten the three-way tensor into a “fat” matrix and exploit the singular subspace geometry based on the matrix SVD for estimation. We establish fine-grained finite-sample entrywise error bounds for all the parameters. Moreover, we develop a novel two-to-infinity singular subspace perturbation theory under arbitrary local dependent noise, which is of independent interest. Simulations and applications to real-world data in genetics, political science, and single-cell sequencing demonstrate the merit of the proposed method.
Bio
Yuqi Gu is an Assistant Professor in the Department of Statistics at Columbia University. She is also a member of the Data Science Institute. Before joining Columbia in 2021, she spent a year as a postdoc at Duke University, mentored by David B. Dunson. In 2020, she received a Ph.D. in Statistics from the University of Michigan, advised by Gongjun Xu. In 2015, she received a B.S. in Mathematics from Tsinghua University.
