Data fusion using factor analysis and low-rank matrix completion

Data fusion involves the integration of multiple related datasets. The statistical file-matching problem is a canonical data fusion problem in multivariate analysis, where the objective is to characterise the joint distribution of a set of variables when only strict subsets of marginal distributions have been observed. Estimation of the covariance matrix of the full set of variables is challenging given the missing-data pattern. Factor analysis models use lower-dimensional latent variables in the data-generating process, and this introduces low-rank components in the complete-data matrix and the population covariance matrix. The low-rank structure of the factor analysis model can be exploited to estimate the full covariance matrix from incomplete data via low-rank matrix completion. We prove the identifiability of the factor analysis model in the statistical file-matching problem under conditions on the number of factors and the number of shared variables over the observed marginal subsets. Additionally, we provide an EM algorithm for parameter estimation. On several real datasets, the factor model gives smaller reconstruction errors in file-matching problems than the common approaches for low-rank matrix completion.