Linked factor analysis

Abstract: 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.