Cover it up! Bipartite graphs uncover identifiability in sparse factor analysis

Factor models are an indispensable tool in dimension reduction in multivariate statistical analysis. Methodological research for factor models is often concerned with identifying rotations that provide the best interpretation of the loadings. This focus on rotational invariance, however, does not ensure unique variance decomposition, which is crucial in many applications where separating common and idiosyncratic variation is key. The present paper provides conditions for variance identification based solely on a counting rule for the binary zero–nonzero pattern of the factor loading matrix which underpins subsequent inference and interpretability. By connecting factor analysis with some classical elements from graph and network theory, it is proven that this condition is sufficient for variance identification without imposing any conditions on the factor loading matrix. An efficient algorithm is designed to verify the seemingly untractable condition in polynomial number of steps. To illustrate the practical relevance of these new insights, the paper makes an explicit connection to post-processing in sparse Bayesian factor analysis. A simulation study and a real world data analysis of financial returns with a time-varying factor model illustrates that verifying variance identification is highly relevant for statistical factor analysis, in particular when the factor dimension is unknown.