An empirical Bayesian strategy for solving the, simultaneous sparse approximation problem
An empirical Bayesian strategy for solving the, simultaneous sparse approximation problem
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DOI：
10.1109/tsp.2007.894265
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发表时间：
2007-07-01
影响因子：
5.4
通讯作者：
Rao, Bhaskar D.
中科院分区：
文献类型：
--
作者：
Wipf, David P.;Rao, Bhaskar D.
给定一个大的过完备基向量字典，目标是使用由共同稀疏性曲线标记的系数展开同时表示L > 1个信号向量。这将标准稀疏表示问题推广到存在多个响应的情况，这些响应是由相同的一小部分特征推定生成的。理想情况下，应该恢复相关的稀疏生成权值，这在许多应用中（例如，源定位）可能具有物理意义。这个问题的一般解是难以解决的，因此，寻求近似程序。基于自动相关性确定的概念，本文使用经验贝叶斯先验估计候选基向量上的方便后验分布。这种特殊的近似执行了一个共同的稀疏轮廓，并始终将其突出的后块放置在同时稀疏恢复所需的权重空间的适当区域上。然后将所得算法与匹配追踪、基础追踪、focus和Jeffreys基于先验的贝叶斯方法的多个响应扩展进行比较，发现它通常优于其他方法。本文还提供了选择这种特殊成本函数的额外动机，包括对全局和局部最小值的分析，以及强调所提出算法与以前方法之间的异同的变分推导。
Given a large overcomplete dictionary of basis vectors, the goal is to simultaneously represent L > 1 signal vectors using coefficient expansions marked by a common sparsity profile. This generalizes the standard sparse representation problem to the case where multiple responses exist that were putatively generated by the same small subset of features. Ideally, the associated sparse generating weights should be recovered, which can have physical significance in many applications (e.g., source localization). The generic solution to this problem is intractable and, therefore, approximate procedures are sought. Based on the concept of automatic relevance determination, this paper uses an empirical Bayesian prior to estimate a convenient posterior distribution over candidate basis vectors. This particular approximation enforces a common sparsity profile and consistently places its prominent posterior mass on the appropriate region of weight-space necessary for simultaneous sparse recovery. The resultant algorithm is then compared with multiple response extensions of matching pursuit, basis pursuit, FOCUSS, and Jeffreys prior-based Bayesian methods, finding that it often outperforms the others. Additional motivation for this particular choice of cost function is also provided, including the analysis of global and local minima and a variational derivation that highlights the similarities and differences between the proposed algorithm and previous approaches.