Fast Bayesian Factor Analysis via Automatic Rotations to Sparsity

Rotational transformations have traditionally played a key role in enhancing the interpretability of factor analysis via post-hoc modifications of the factor model orientation. Regularization methods also serve to achieve this goal by prioritizing sparse loading matrices. In this work, we cross-fertilize these two paradigms within a unifying Bayesian framework. Our approach deploys intermediate factor rotations throughout the learning process, greatly enhancing the effectiveness of sparsity inducing priors. These automatic rotations to sparsity are embedded within a PXL-EM algorithm, a Bayesian variant of parameter-expanded EM for posterior mode detection. By iterating between soft-thresholding of small factor loadings and transformations of the factor basis, we obtain (a) dramatic accelerations, (b) robustness against poor initializations and (c) better oriented sparse solutions. For accurate recovery of factor loadings, we deploy a two-component refinement of the Laplace prior, the spike-and-slab LASSO prior. This prior is coupled with the Indian Buffet Process (IBP) prior to avoid the pre-specification of the factor cardinality. The ambient dimensionality is learned from the posterior, which is shown to reward sparse matrices. Our deployment of PXL-EM performs a dynamic posterior exploration, outputting a solution path indexed by a sequence of spike-and-slab priors. A companion criterion, motivated as an integral lower bound, is provided to effectively select the best recovery. The potential of the proposed procedure is demonstrated on both simulated and real high-dimensional data, which would render posterior simulation impractical.