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Factor analysis provides a canonical framework for imposing lower-dimensionalstructure such as sparse covariance in high-dimensional data. High-dimensionaldata on the same set of variables are often collected under differentconditions, for instance in reproducing studies across research groups. In suchcases, it is natural to seek to learn the shared versus condition-specificstructure. Existing hierarchical extensions of factor analysis have beenproposed, but face practical issues including identifiability problems. Toaddress these shortcomings, we propose a class of SUbspace Factor Analysis(SUFA) models, which characterize variation across groups at the level of alower-dimensional subspace. We prove that the proposed class of SUFA modelslead to identifiability of the shared versus group-specific components of thecovariance, and study their posterior contraction properties. Taking a Bayesianapproach, these contributions are developed alongside efficient posteriorcomputation algorithms. Our sampler fully integrates out latent variables, iseasily parallelizable and has complexity that does not depend on sample size.We illustrate the methods through application to integration of multiple geneexpression datasets relevant to immunology.

Inferring Covariance Structure from Multiple Data Sources via Subspace Factor Analysis