Bayesian group latent factor analysis with structured sparsity

Latent factor models are the canonical statistical tool for exploratory analyses of lowdimensional linear structure for an observation matrix with p features across n samples. We develop a Bayesian group factor analysis (BGFA) model that extends the factor model to multiple coupled observation matrices. Our model puts a structured Bayesian hierarchical prior on the joint factor loading matrix, which achieves shrinkage effect at both a local level (element-wise shrinkage) and a factor level (column-wise shrinkage) with non-parametric behavior that removes unnecessary factors. With two observations, our model reduces to Bayesian canonical correlation analysis (BCCA). We exploit the shrinkage behavior in the BGFA model to recover covariance structure across all subsets of the observation matrices where this signal exists. We validate our model on simulated data with substantial structure and compare recovered factor loadings against results from related methods. We then show the results of applying BGFA to two genomics studies for different analytic aims: identifying gene co-expression networks specific to one of two conditions, and recovering sets of genetic variants that jointly regulate transcription of a collection of genes. We illustrate the unique ability of BGFA to use multiple observations of the same samples to guide linear projection of the data onto a latent space, producing meaningful and robust lowdimensional representations, as compared with ‘unsupervised’ projections from traditional factor analysis or principal components analysis.