Regularization is a common tool in variational inverse problems to impose assumptions on the parameters of the problem. One such assumption is sparsity, which is commonly promoted using lasso and total variation-like regularization. Although the solutions to many such regularized inverse problems can be considered as points of maximum probability of well-chosen posterior distributions, samples from these distributions are generally not sparse. In this paper, we present a framework for implicitly defining a probability distribution that combines the effects of sparsity imposing regularization with Gaussian distributions. Unlike continuous distributions, these implicit distributions can assign positive probability to sparse vectors. We study these regularized distributions for various regularization functions including total variation regularization and piecewise linear convex functions. We apply the developed theory to uncertainty quantification for Bayesian linear inverse problems and derive a Gibbs sampler for a Bayesian hierarchical model. To illustrate the difference between our sparsity-inducing framework and continuous distributions, we apply our framework to small-scale deblurring and computed tomography examples.

翻译：正则化是变分反问题中用于对问题参数施加假设的常用工具。其中一种常见假设是稀疏性，通常通过套索（lasso）和全变分（total variation）类正则化来促进。尽管许多此类正则化反问题的解可被视为精心选择的后验分布的概率最大点，但这些分布的样本通常并不稀疏。本文提出一个框架，用于隐式定义一种结合稀疏性诱导正则化与高斯分布效应的概率分布。与连续分布不同，这些隐式分布能够对稀疏向量赋予正概率。我们研究了针对各种正则化函数（包括全变分正则化和分段线性凸函数）的这些正则化分布。我们将所发展的理论应用于贝叶斯线性反问题的不确定性量化，并推导出贝叶斯分层模型的吉布斯采样器。为阐明我们提出的稀疏性诱导框架与连续分布之间的差异，我们将该框架应用于小规模去模糊和计算机断层扫描实例。