Identification of Nonlinear and Dependent Latent Factor Structure through Clique Search

Learning the structure of latent factor models involves two central challenges: (1) estimating the number of latent factors and (2) learning the support of the mapping from latent variables to observed variables. This is especially challenging for nonparametric regimes and nonlinear settings. We propose a method for latent structure learning, based on pairwise dependence measures on the observed variables using a graph-theoretic representation. We show that both the number of latent factors and nonlinear mapping structure can be identified from the distribution of observed variables under mild structural assumptions. Unlike prior work restricted to linear correlations, we establish identifiability and consistency for a general class of dependence measures under nonlinear factor models. This motivates a Dependence Thresholding (DT) algorithm, which jointly estimates the number of latent factors and nonlinear mapping structure from observational data alone. We pair this with a neural network architecture constrained by the nonlinear mapping structure, to recover the nonlinear function. Through simulation studies, we show that the DT algorithm is accurate in practice, even when using flexible methods such as neural networks, and exhibits robustness against violations of its assumptions in high-dimensional settings.