  Factor models are widely used in the analysis of high-dimensional data in
several fields of research. Estimating a factor model, in particular its
covariance matrix, from partially observed data vectors is very challenging. In
this work, we show that when the data are structurally incomplete, the factor
model likelihood function can be decomposed into the product of the likelihood
functions of multiple partial factor models relative to different subsets of
data. If these multiple partial factor models are linked together by common
parameters, then we can obtain complete maximum likelihood estimates of the
factor model parameters and thereby the full covariance matrix. We call this
framework Linked Factor Analysis (LINFA). LINFA can be used for covariance
matrix completion, dimension reduction, data completion, and graphical
dependence structure recovery. We propose an efficient Expectation-Maximization
algorithm for maximum likelihood estimation, accelerated by a novel group
vertex tessellation (GVT) algorithm which identifies a minimal partition of the
vertex set to implement an efficient optimization in the maximization steps. We
illustrate our approach in an extensive simulation study and in the analysis of
calcium imaging data collected from mouse visual cortex.
