Covariance-Free Sparse Bayesian Learning

TL;DR

This paper introduces CoFEM, a covariance-free method for sparse Bayesian learning that significantly accelerates inference in high-dimensional problems by avoiding large covariance matrix computations.

Contribution

The paper presents CoFEM, a novel covariance-free inference algorithm for SBL that improves scalability and speed without compromising accuracy.

Findings

CoFEM is up to thousands of times faster than existing methods.

CoFEM maintains high coding accuracy in high-dimensional problems.

Applications demonstrate CoFEM's practicality in calcium imaging and MRI reconstruction.

Abstract

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.