Stability of the elastic net estimator

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• 作者：Yi Shen^a ; ^b ; ^c ; ^{yshen@zstu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Bin Han^b ; ^{bhan@ualberta.ca" class="auth_mail" title="E-mail the corresponding author} ; ; Elena Braverman^c ; ^{maelena@math.ucalgary.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Compressed sensing ; Elastic net ; Stable recovery ; Correlated random matrix ; Grouping effect
• 刊名：Journal of Complexity
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：32
• 期：1
• 页码：20-39
• 全文大小：637 K

文摘

The elastic net is a regularized least squares regression method that has been widely used in learning and variable selection. The elastic net regularization linearly combines an l₁$l_1$ penalty term (like the lasso) and an l₂$l_2$ penalty term (like ridge regression). The l₁$l_1$ penalty term enforces sparsity of the elastic net estimator, whereas the l₂$l_2$ penalty term ensures democracy among groups of correlated variables. Compressed sensing is currently an extensively studied technique for efficiently reconstructing a sparse vector from much fewer samples/observations. In this paper we study the elastic net in the setting of sparse vector recovery. For recovering sparse vectors from few observations by employing the elastic net regression, we prove in this paper that the elastic net estimator is stable provided that the underlying measurement/design matrix satisfies the commonly required restricted isometry property or the sparse approximation property. It is well known that many independent random measurement matrices satisfy the restricted isometry property while random measurement matrices generated by highly correlated Gaussian random variables satisfy the sparse approximation property. As a byproduct, we establish a uniform bound for the grouping effect of the elastic net. Some numerical experiments are provided to illustrate our theoretical results on stability and grouping effect of the elastic net estimator.