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Proximal linearized iteratively reweighted least squares for a class of nonconvex and nonsmooth problems

机译:近似线性化迭代重加权最小二乘法   非凸面和非光滑的问题

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摘要

For solving a wide class of nonconvex and nonsmooth problems, we propose aproximal linearized iteratively reweighted least squares (PL-IRLS) algorithm.We first approximate the original problem by smoothing methods, and secondwrite the approximated problem into an auxiliary problem by introducing newvariables. PL-IRLS is then built on solving the auxiliary problem by utilizingthe proximal linearization technique and the iteratively reweighted leastsquares (IRLS) method, and has remarkable computation advantages. We show thatPL-IRLS can be extended to solve more general nonconvex and nonsmooth problemsvia adjusting generalized parameters, and also to solve nonconvex and nonsmoothproblems with two or more blocks of variables. Theoretically, with the help ofthe Kurdyka- Lojasiewicz property, we prove that each bounded sequencegenerated by PL-IRLS globally converges to a critical point of the approximatedproblem. To the best of our knowledge, this is the first global convergenceresult of applying IRLS idea to solve nonconvex and nonsmooth problems. Atlast, we apply PL-IRLS to solve three representative nonconvex and nonsmoothproblems in sparse signal recovery and low-rank matrix recovery and obtain newglobally convergent algorithms.
机译:为了解决一类非凸和非光滑问题,我们提出了一种近似线性化迭代最小加权平方(PL-IRLS)算法。首先通过平滑方法逼近原始问题,然后通过引入新变量将逼近问题写成一个辅助问题。然后,利用近端线性化技术和迭代加权最小二乘法(IRLS),在解决辅助问题的基础上构建PL-IRLS,具有显着的计算优势。我们表明,PL-IRLS可以扩展为通过调整广义参数来解决更一般的非凸和非光滑问题,还可以解决带有两个或多个变量块的非凸和非光滑问题。从理论上讲,借助Kurdyka-Lojasiewicz属性,我们证明了PL-IRLS生成的每个有界序列都全局收敛到近似问题的临界点。据我们所知,这是应用IRLS思想解决非凸和非光滑问题的第一个全球收敛结果。最后,我们应用PL-IRLS解决了稀疏信号恢复和低秩矩阵恢复中的三个代表性非凸和非光滑问题,并获得了新的全局收敛算法。

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