Bayesian rank penalization

Abstract Rank minimization is a key component of many computer vision and machine learning methods, including robust principal component analysis (RPCA) and low-rank representations (LRR). However, usual methods rely on optimization to produce a point estimate without characterizing uncertainty in this estimate, and also face difficulties in tuning parameter choice. Both of these limitations are potentially overcome with Bayesian methods, but there is currently a lack of general purpose Bayesian approaches for rank penalization. We address this gap using a positive generalized double Pareto prior, illustrating the approach in RPCA and LRR. Posterior computation relies on hybrid Gibbs sampling and geodesic Monte Carlo algorithms. We assess performance in simulation examples, and benchmark data sets.