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May 2021 A Unified Primal Dual Active Set Algorithm for Nonconvex Sparse Recovery
Jian Huang, Yuling Jiao, Bangti Jin, Jin Liu, Xiliang Lu, Can Yang
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Statist. Sci. 36(2): 215-238 (May 2021). DOI: 10.1214/19-STS758


In this paper, we consider the problem of recovering a sparse signal based on penalized least squares formulations. We develop a novel algorithm of primal-dual active set type for a class of nonconvex sparsity-promoting penalties, including 0, bridge, smoothly clipped absolute deviation, capped 1 and minimax concavity penalty. First, we establish the existence of a global minimizer for the related optimization problems. Then we derive a novel necessary optimality condition for the global minimizer using the associated thresholding operator. The solutions to the optimality system are coordinatewise minimizers, and under minor conditions, they are also local minimizers. Upon introducing the dual variable, the active set can be determined using the primal and dual variables together. Further, this relation lends itself to an iterative algorithm of active set type which at each step involves first updating the primal variable only on the active set and then updating the dual variable explicitly. When combined with a continuation strategy on the regularization parameter, the primal dual active set method is shown to converge globally to the underlying regression target under certain regularity conditions. Extensive numerical experiments with both simulated and real data demonstrate its superior performance in terms of computational efficiency and recovery accuracy compared with the existing sparse recovery methods.


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Jian Huang. Yuling Jiao. Bangti Jin. Jin Liu. Xiliang Lu. Can Yang. "A Unified Primal Dual Active Set Algorithm for Nonconvex Sparse Recovery." Statist. Sci. 36 (2) 215 - 238, May 2021.


Published: May 2021
First available in Project Euclid: 19 April 2021

Digital Object Identifier: 10.1214/19-STS758

Keywords: consistency , continuation , nonconvex penalty , primal-dual active set algorithm , Sparsity

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.36 • No. 2 • May 2021
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