低秩矩阵优化若干新进展

低秩矩阵优化是一类含有秩极小或秩约束的矩阵优化问题,在统计与机器学习、信号与图像处理、通信与量子计算、系统识别与控制、经济与金融等众多学科领域有着广泛应用,是当前最优化及其相关领域的一个重点研究方向.然而,低秩矩阵优化是一个NP-难的非凸非光滑优化问题,其研究成果并非十分丰富,亟待进一步深入研究.主要从理论和算法两个方面总结和评述若干新结果,同时列出相关的重要文献,奉献给读者.

Some advances in low-rank matrix optimization

Low-rank matrix optimization is a class of matrix optimization problems with rank minimization or rank constraint.With wide applications ranging from statistics and machine learning,signal and image processing,communication and quantum computing,system identification and control,to economics and finance,low-rank matrix optimization is currently a key research direction in optimization and related fields.However,due to the intrinsic non-convexity and discontinuity in the rank function,low-rank matrix optimization is generally NP-hard.Existing research results in this direction are not very rich,and further research is urgently needed.In this paper,we mainly summarize and review some latest research results on low-rank matrix optimization in theory and in algorithm,along with related important references,so as to dedicate to readers.