Research articles
ScienceAsia 44 (2018): 311318 doi:
10.2306/scienceasia15131874.2018.44.311
An Inverse MatrixFree Proximal Point Algorithm for Compressive Sensing
Hongchun Sun^{a,*}, Min Sun^{b,c}, Bohan Zhang^{d}
ABSTRACT: In recent years, the compressive sensing (CS) has received considerable attention in signal processing
and statistical inference. The classical proximal point algorithm (PPA) for some reformulations of CS often involves
an inverse matrix at each iteration, which usually requires expensive computation if high dimensional variables are
considered. Our contribution in this paper is to propose a novelly inverse matrixfree PPA to solve CS for the first time.
More specifically, we first establish some equivalent reformulations of CS, which are smooth and convex. Based on
these equivalent reformations, a new proximal point algorithm is proposed to solve CS, whose inverse matrix can be
removed by choosing some special parameter. Thus we get an inverse matrixfree PPA, which is implementable for large
scale CS. Global convergence of the new PPA and its inverse matrixfree version is established. Comparative numerical
results are presented, which substantiate the efficacy and validity of the inverse matrixfree PPA for solving some sparse
signal recovery problems.
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^{a} 
School of Mathematics and Statistics, Linyi University, Shandong, Linyi, 276005, P. R. China 
^{b} 
School of Mathematics and Statistics, Zaozhuang University, Shandong, Zaozhuang, 277160, P. R. China 
^{c} 
School of Management, Qufu Normal University, Shandong, Rizhao, 276826, P. R. China 
^{d} 
School of Information Science and Engineering, Jinan University, Shandong, Jinan, 250022, P. R. China 
* Corresponding author, Email: hcsun68@126.com
Received 1 Jun 2018, Accepted 26 Aug 2018
