MINIMIZATION OF L1 OVER L2 FOR SPARSE SIGNAL RECOVERY WITH CONVERGENCE GUARANTEE

报告人简介：陶敏，女，南京大学数学系教授，博士生导师，主要从事一阶算法的理论及其应用的研究。本、硕、博毕业于南京大学数学系。2013评为江苏省优秀博士生。2012.9-2013.9，在香港中文大学数学系从事博士后工作。2016.3-2017.5，在美国南加州大学做访问学者。成果发表在 SIAM J. Optim.，SIAM J. Sci. Comput.，Math. Oper. Res.，SIAM J. Imaging Sci., Math. Comp.，J. Sci. Comput.等优化领重要期刊上。有数篇高被引文章，1篇文章入围中国科学领域热点论文。2017年荣获世界华人数学家大会杰出论文奖。主持国家、江苏省面上基金项目数项，以及参加国家重点研发计划。

The ratio of the L1 and L2 norms, denoted by L1/L2, becomes attractive due to its scale-invariant property when approximating the L0 norm to promote sparsity. In this paper, we incorporate the L1/L2 formalism into an unconstrained model in order to deal with both noiseless and noisy observations. To design an efficient algorithm, we derive an analytical solution for the proximal operator of the L1/L2 functional. Since the analytical solution depends on the sparsity of an unknown signal, we develop a bisection search method to find the desired sparsity and the corresponding solution to the proximal operator of L1/L2. With the newly developed solver of the proximal operator, we propose a specific variable-splitting scheme for the alternating direction method of multipliers so that we can establish its global convergence under mild assumptions and prove its linear convergence rate under suitable conditions. Experimentally, we conduct extensive numerical simulations to demonstrate the efficiency of the proposed approach over the state-of-the-art methods in sparse signal recovery with and without noise.