Fast L1–L2 Minimization via a Proximal Operator
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This paper aims to develop new and fast algorithms for recovering a sparse vector from a small number of measurements, which is a fundamental problem in the field of compressive sensing (CS). Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, and conventional methods such as \(L_1\) minimization do not work well. Recently, the difference of the \(L_1\) and \(L_2\) norms, denoted as \(L_1\)–\(L_2\), is shown to have superior performance over the classic \(L_1\) method, but it is computationally expensive. We derive an analytical solution for the proximal operator of the \(L_1\)–\(L_2\) metric, and it makes some fast \(L_1\) solvers such as forward–backward splitting (FBS) and alternating direction method of multipliers (ADMM) applicable for \(L_1\)–\(L_2\). We describe in details how to incorporate the proximal operator into FBS and ADMM and show that the resulting algorithms are convergent under mild conditions. Both algorithms are shown to be much more efficient than the original implementation of \(L_1\)–\(L_2\) based on a difference-of-convex approach in the numerical experiments.
KeywordsCompressive sensing Proximal operator Forward–backward splitting Alternating direction method of multipliers Difference-of-convex
Mathematics Subject Classification90C26 65K10 49M29
The authors would like to thank Zhi Li and the anonymous reviewers for valuable comments.
- 5.Chartrand, R., Yin, W.: Iteratively reweighted algorithms for compressive sensing. In: International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 3869–3872 (2008)Google Scholar
- 13.Krishnan, D., Fergus, R.: Fast image deconvolution using hyper-Laplacian priors. In: Advances in Neural Information Processing Systems (NIPS), pp. 1033–1041 (2009)Google Scholar
- 16.Li, H., Lin, Z.: Accelerated proximal gradient methods for nonconvex programming. In: Advances in Neural Information Processing Systems, pp. 379–387 (2015)Google Scholar
- 19.Lou, Y., Osher, S., Xin, J.: Computational aspects of l1-l2 minimization for compressive sensing. In: Le Thi, H., Pham Dinh, T., Nguyen, N. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences. Advances in Intelligent Systems and Computing, vol. 359, pp. 169–180. Springer, Cham (2015)Google Scholar
- 30.Wang, Y., Yin, W., Zeng, J.: Global convergence of ADMM in nonconvex nonsmooth optimization. arXiv:1511.06324 [cs, math] (2015)
- 35.Zhang, S., Xin, J.: Minimization of transformed \(l_1\) penalty: Theory, difference of convex function algorithm, and robust application in compressed sensing. arXiv preprint arXiv:1411.5735 (2014)