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.

中文翻译：

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无协方差稀疏贝叶斯学习


稀疏贝叶斯学习 (SBL) 是一个强大的框架，用于解决稀疏编码问题，同时还提供不确定性量化。由于需要维护较大的协方差矩阵，最流行的 SBL 推理算法对于高维问题表现出极高的计算成本。为了解决这个问题，我们引入了一种加速 SBL 推理的新方法——称为无协方差期望最大化（CoFEM）——它避免了协方差矩阵的显式计算。 CoFEM 求解多个线性系统以获得 SBL 所需的后验统计的无偏估计。这是通过利用数值线性代数的创新来实现的，例如预处理共轭梯度和鲜为人知的对角线估计规则。对于一大类压缩感知矩阵，我们提供了理论依据，说明为什么我们的方法在高维设置中具有良好的扩展性。通过模拟，我们表明 CoFEM 的速度比现有基线快数千倍，而不会牺牲编码精度。通过在钙成像反卷积和多对比 MRI 重建中的应用，我们表明 CoFEM 使 SBL 能够轻松解决实际感兴趣的高维稀疏编码问题。