Provable Guarantees for Estimating Covariances between Latent Variables with Application to Precision Matrix Estimation

In many scientific fields, key variables of interest are latent---either because they cannot be measured directly or because doing so is prohibitively expensive. As a result, researchers often rely on high-dimensional surrogate observations and must infer relationships among the unobserved quantities. In this work, we address a fundamental challenge: How can one estimate the covariance between variables that are not directly observable? We consider a model where each latent variable elicits high-dimensional observable covariates. Under our model, we propose a method that estimates several spiked covariances from the observed variables and then reconstructs the covariance matrix among the latent variables. Our estimator achieves quadratic-time complexity with respect to the number of latent variables and only requires the sample size to be logarithmic in the number of latent variables. As an immediate application, our procedure can be leveraged to recover the conditional independence structure among the latent variables, providing interpretable insights. Extensive synthetic experiments validate our theory, demonstrating accurate estimation in practice.