稀疏低秩矩阵优化模型及其在语音去噪中的应用  

Sparse low-rank matrix optimization model with application in speech denoising

作　　者：马婷婷 刘潇 沈春根 薛文娟 MA Ting-ting;LIU Xiao;SHEN Chun-gen;XUE Wen-juan(College of Science,University of Shanghai for Science and Technology,Shanghai 200093,China;School of Mathematics and Physics,Shanghai University of Electric Power,Shanghai 200090,China)

机构地区：[1]上海理工大学理学院,上海200093 [2]上海电力大学数理学院,上海200090

出　　处：《高校应用数学学报（A辑）》2022年第2期127-141,共15页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：国家自然科学基金(11601318)。

摘　　要：为了更好地恢复低秩稀疏结构的矩阵,提出了一种改进的稀疏低秩矩阵优化模型.不同于单纯地在目标中加入核范数和l_(1)范数来保证目标矩阵的低秩性和稀疏性,该模型一是通过加入稀疏性约束以更加严格地保证了目标矩阵的稀疏性,二是通过加入线性约束以刻画目标矩阵中可能存在的先验信息.为求解该非凸非光滑模型,首先利用精确罚函数将原模型转化为一个约束DC(两个凸函数之差)优化模型,其次将其线性化为凸子问题,并用交替方向乘子法求解该子问题.在一定的假设条件下,该算法具有全局收敛性.数值实验表明该模型在语音去噪问题上具有良好的效果.This paper proposes a sparse low-rank matrix optimization model to recover a matrix with sparse low-rank structure.Different from simply adding the nuclear norm and norm to the objective function to promote the low-rankness and sparsity of the target matrix,this model not only guarantees the sparsity of the target matrix by adding the sparsity constraint,but also takes advantage of the possible prior information in the target matrix via some linear constraints.To solve the non-convex and non-smooth model,the original model is first transformed into a constrained DC(difference between two convex functions)optimization model using the exact penalty function,and then it is linearized into a convex subproblem which is solved by the alternating direction multiplier method.Under some assumptions,the algorithm has global convergence.Numerical experiments show that the model has a good effect on speech denoising.

关 键 词：低秩 稀疏 两个凸函数之差 交替方向乘子法 

分 类 号：O221.2[理学—运筹学与控制论]