We study linear models under heavy-tailed priors from a probabilistic viewpoint. Instead of computing a single sparse most probable (MAP) solution as in standard compressed sensing, the focus in the Bayesian framework shifts towards capturing the full posterior distribution on the latent variables, which allows quantifying the estimation uncertainty and learning model parameters using maximum likelihood. The exact posterior distribution under the sparse linear model is intractable and we concentrate on a number of alternative variational Bayesian techniques to approximate it. Repeatedly computing Gaussian variances turns out to be a key requisite for all these approximations and constitutes the main computational bottleneck in applying variational techniques in large-scale problems. We leverage on the recently proposed Perturb-and-MAP algorithm for drawing exact samples from Gaussian Markov random fields (GMRF). The main technical contribution of our paper is to show that estimating Gaussian variances using a relatively small number of such efficiently drawn random samples is much more effective than alternative general-purpose variance estimation techniques. Interestingly, this instills a stochastic flavor into the otherwise deterministic variational framework. By reducing the problem of variance estimation to standard optimization primitives, the proposed variational algorithms are fully scalable and parallelizable, allowing Bayesian computations in extremely large-scale problems with the same memory and time complexity requirements as conventional point estimation techniques. We illustrate these ideas with experiments in image deblurring.

本研究使用贝叶斯压缩感知框架从概率学的角度研究了重尾先验下的线性模型，并借助基于随机场的 Perturb-and-MAP 算法提出了一种高效的方法近似估计高斯方差，实现对完整后验分布的捕捉及模型参数的学习并通过实验在图像去模糊中得到了良好效果。

大规模贝叶斯压缩感知中的高效变分推断