Bayesian group latent factor analysis with structured sparsity
This work addresses the need for flexible and scalable factor analysis in high-dimensional data like genomics and document analysis, though it is incremental as it builds on existing Bayesian and factor model frameworks.
The authors tackled the problem of extending latent factor models to multiple coupled observation matrices, such as in canonical correlation analysis, by developing a structured Bayesian group factor analysis model with a prior that encourages element-wise and column-wise shrinkage. The result is a method that recovers sparse signals in the presence of dense effects, scales to large datasets, and performs competitively against state-of-the-art approaches on simulated and real data.
Latent factor models are the canonical statistical tool for exploratory analyses of low-dimensional linear structure for an observation matrix with p features across n samples. We develop a structured Bayesian group factor analysis model that extends the factor model to multiple coupled observation matrices; in the case of two observations, this reduces to a Bayesian model of canonical correlation analysis. The main contribution of this work is to carefully define a structured Bayesian prior that encourages both element-wise and column-wise shrinkage and leads to desirable behavior on high-dimensional data. In particular, our model puts a structured prior on the joint factor loading matrix, regularizing at three levels, which enables element-wise sparsity and unsupervised recovery of latent factors corresponding to structured variance across arbitrary subsets of the observations. In addition, our structured prior allows for both dense and sparse latent factors so that covariation among either all features or only a subset of features can both be recovered. We use fast parameter-expanded expectation-maximization for parameter estimation in this model. We validate our method on both simulated data with substantial structure and real data, comparing against a number of state-of-the-art approaches. These results illustrate useful properties of our model, including i) recovering sparse signal in the presence of dense effects; ii) the ability to scale naturally to large numbers of observations; iii) flexible observation- and factor-specific regularization to recover factors with a wide variety of sparsity levels and percentage of variance explained; and iv) tractable inference that scales to modern genomic and document data sizes.