Sparse recovery via differential inclusions

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• 作者：Stanley Osher^a ; ^{sjo@math.ucla.edu" class="auth_mail" title="E-mail the corresponding author} ; Feng Ruan^b ; ^c ; ^{fengruan@stanford.edu" class="auth_mail" title="E-mail the corresponding author} ; Jiechao Xiong^b ; ^{xiongjiechao@pku.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Yuan Yao^b ; ^{yuany@math.pku.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Wotao Yin^a ; ^{wotaoyin@math.ucla.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Linearized Bregman ; Differential inclusion ; Early stopping regularization ; Statistical consistency
• 刊名：Applied and Computational Harmonic Analysis
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：41
• 期：2
• 页码：436-469
• 全文大小：1652 K

文摘

In this paper, we recover sparse signals from their noisy linear measurements by solving nonlinear differential inclusions, which is based on the notion of inverse scale space (ISS) developed in applied mathematics. Our goal here is to bring this idea to address a challenging problem in statistics, i.e. finding the oracle estimator which is unbiased and sign consistent using dynamics. We call our dynamics Bregman ISS and Linearized Bregman ISS. A well-known shortcoming of LASSO and any convex regularization approaches lies in the bias of estimators. However, we show that under proper conditions, there exists a bias-free and sign-consistent point on the solution paths of such dynamics, which corresponds to a signal that is the unbiased estimate of the true signal and whose entries have the same signs as those of the true signs, i.e. the oracle estimator. Therefore, their solution paths are regularization paths better than the LASSO regularization path, since the points on the latter path are biased when sign-consistency is reached. We also show how to efficiently compute their solution paths in both continuous and discretized settings: the full solution paths can be exactly computed piece by piece, and a discretization leads to Linearized Bregman iteration  , which is a simple iterative thresholding rule and easy to parallelize. Theoretical guarantees such as sign-consistency and minimax optimal l₂$l_2$-error bounds are established in both continuous and discrete settings for specific points on the paths. Early-stopping rules for identifying these points are given. The key treatment relies on the development of differential inequalities for differential inclusions and their discretizations, which extends the previous results and leads to exponentially fast recovering of sparse signals before selecting wrong ones.