A common task in inverse problems and imaging is finding a solution that is sparse, in the sense that most of its components vanish. In the framework of compressed sensing, general results guaranteeing exact recovery have been proven. In practice, sparse solutions are often computed combining $\ell_1$-penalized least squares optimization with an appropriate numerical scheme to accomplish the task. A computationally efficient alternative for finding sparse solutions to linear inverse problems is provided by Bayesian hierarchical models, in which the sparsity is encoded by defining a conditionally Gaussian prior model with the prior parameter obeying a generalized gamma distribution. An iterative alternating sequential (IAS) algorithm has been demonstrated to lead to a computationally efficient scheme, and combined with Krylov subspace iterations with an early termination condition, the approach is particularly well suited for large scale problems. Here the Bayesian approach to sparsity is extended to problems whose solution allows a sparse coding in an overcomplete system such as composite frames. It is shown that among the multiple possible representations of the unknown, the IAS algorithm, and in particular, a hybrid version of it, is effectively identifying the most sparse solution. Computed examples show that the method is particularly well suited not only for traditional imaging applications but also for dictionary learning problems in the framework of machine learning.
翻译:反向问题和成像的共同任务是找到一种稀少的解决方案,即其大部分组成部分消失。在压缩遥感的框架内,保证准确恢复的一般结果已经得到证明。在实践中,往往计算出稀有的解决方案,将美元=1美元=1美元=1美元=1美元=平方平方平方平优化与适当的数字方案相结合,以完成这项任务。巴伊西亚等级模型为寻找线性反问题的稀疏解决方案提供了一种计算高效的替代方法,在巴伊西亚等级模型中,通过界定一个有条件的Gaussian先前模型,以先前的参数满足普遍伽马分布,来编码松散。一个迭代交替顺序算法(IAS)已被证明可导致一个计算效率高计划,并与Krylov子空格和早期终止条件相结合,这种方法特别适合大规模问题。这里,巴伊西亚对宽阔度方法的延伸至问题,其解决办法允许在一个过于完善的系统,例如综合框架中进行稀薄的编码。在多种可能的表达方式中,IAS算法,特别是混合式的混合式序列算法的算法也有效确定了最缺乏的模型。