Two new approaches to compressed sensing exhibiting both robust sparse recovery and the grouping effect
Two new approaches to compressed sensing exhibiting both robust sparse recovery and the grouping effect
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两种新的压缩感知方法同时表现出鲁棒的稀疏恢复和分组效应
DOI：
10.1109/indiancc.2017.7846482
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
M. Vidyasagar
中科院分区：
文献类型：
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作者：
M. Ahsen;Niharika Challapalli;M. Vidyasagar
本文介绍了一种新的用于稀疏回归和压缩感知的优化公式，称为CLOT (Combined L-One and Two)，其中正则化器是1-和2-范数的凸组合。这个公式不同于弹性网（EN）公式，在弹性网中，正则化器是1-和2-范数平方的凸组合。这个看似简单的修改有相当重要的后果。特别是，本文表明，在压缩感知的背景下，EN公式不能实现稀疏向量的鲁棒恢复，而新的CLOT公式可以。此外，与EN但与LASSO不同的是，CLOT公式实现了分组效果，其中测量（或设计）矩阵的高度相关列的系数被赋予大致可比较的值。值得注意的是，LASSO没有分组效果，EN（如图所示）没有实现鲁棒稀疏恢复。因此，CLOT公式结合了LASSO（鲁棒稀疏恢复）和EN（分组效果）的最佳特征。CLOT公式是另一种称为SGL（稀疏组LASSO）的特殊情况，该公式先前在文献中介绍过，但没有对分组效果或鲁棒稀疏恢复进行任何分析。这里显示，SGL实现了鲁棒稀疏恢复，并且还实现了分组效应的一个版本，如果列属于同一组，则测量（或设计）矩阵的高度相关列的系数被赋予大致可比较的值。
In this paper we introduce a new optimization formulation for sparse regression and compressed sensing, called CLOT (Combined L-One and Two), wherein the regularizer is a convex combination of the ℓ1- and ℓ2-norms. This formulation differs from the Elastic Net (EN) formulation, in which the regularizer is a convex combination of the ℓ1- and ℓ2-norm squared. This seemingly simple modification has fairly significant consequences. In particular, it is shown in this paper that the EN formulation does not achieve robust recovery of sparse vectors in the context of compressed sensing, whereas the new CLOT formulation does so. Also, like EN but unlike LASSO, the CLOT formulation achieves the grouping effect, wherein coefficients of highly correlated columns of the measurement (or design) matrix are assigned roughly comparable values. It is noteworthy that LASSO does not have the grouping effect and EN (as shown here) does not achieve robust sparse recovery. Therefore the CLOT formulation combines the best features of both LASSO (robust sparse recovery) and EN (grouping effect). The CLOT formulation is a special case of another one called SGL (Sparse Group LASSO) which was introduced into the literature previously, but without any analysis of either the grouping effect or robust sparse recovery. It is shown here that SGL achieves robust sparse recovery, and also achieves a version of the grouping effect in that coefficients of highly correlated columns of the measurement (or design) matrix are assigned roughly comparable values, if the columns belong to the same group.