Covariance-Free Sparse Bayesian Learning
This work addresses a computational bottleneck for researchers and practitioners using SBL in high-dimensional sparse coding problems, such as in medical imaging, by making it tractable.
The paper tackles the high computational cost of sparse Bayesian learning (SBL) in high-dimensional problems by introducing covariance-free expectation maximization (CoFEM), which avoids explicit covariance matrix computation. The result is a method that can be up to thousands of times faster than existing baselines while maintaining coding accuracy, as demonstrated in simulations and applications like calcium imaging and MRI reconstruction.
Sparse Bayesian learning (SBL) is a powerful framework for tackling the sparse coding problem while also providing uncertainty quantification. The most popular inference algorithms for SBL exhibit prohibitively large computational costs for high-dimensional problems due to the need to maintain a large covariance matrix. To resolve this issue, we introduce a new method for accelerating SBL inference -- named covariance-free expectation maximization (CoFEM) -- that avoids explicit computation of the covariance matrix. CoFEM solves multiple linear systems to obtain unbiased estimates of the posterior statistics needed by SBL. This is accomplished by exploiting innovations from numerical linear algebra such as preconditioned conjugate gradient and a little-known diagonal estimation rule. For a large class of compressed sensing matrices, we provide theoretical justifications for why our method scales well in high-dimensional settings. Through simulations, we show that CoFEM can be up to thousands of times faster than existing baselines without sacrificing coding accuracy. Through applications to calcium imaging deconvolution and multi-contrast MRI reconstruction, we show that CoFEM enables SBL to tractably tackle high-dimensional sparse coding problems of practical interest.