Covariance-Free Sparse Bayesian Learning

Sparse Bayesian learning (SBL) is a powerful framework for tackling the sparse coding problem while also providing uncertainty quantification. However, the most popular inference algorithms for SBL become too expensive for high-dimensional problems due to the need to maintain a large covariance matrix. To resolve this issue, we introduce a new SBL inference algorithm that avoids explicit computation of the covariance matrix, thereby saving significant time and space. Instead of performing costly matrix inversions, our covariance-free method solves multiple linear systems to obtain provably unbiased estimates of the posterior statistics needed by SBL. These systems can be solved in parallel, enabling further acceleration of the algorithm via graphics processing units. In practice, our method can be up to thousands of times faster than existing baselines, reducing hours of computation time to seconds. We showcase how our new algorithm enables SBL to tractably tackle high-dimensional signal recovery problems, such as deconvolution of calcium imaging data and multi-contrast reconstruction of magnetic resonance images. Finally, we open-source a toolbox containing all of our implementations to drive future research in SBL.