The Augmented Lagrangian Multiplier Method For Nonconvex Regular Matrix Regression

With the development of modern science and technology and the arrival of the era of big data,matrix regression problems are widely used in scientific research and prac-tical applications,such as machine learning and artificial intelligence,gene expression analysis,neural networks,medical imaging,disease diagnosis and Treatment,risk man-agement,etc.Mixed data in both matrix and vector forms has also attracted widespread attention.There are significant advantages in variable selection.Given the ultrahigh di-mensionality and the complex structure,which contains matrices and vectors,the Ma-trix Regression Minimization becomes crucial for the analysis of those data.Recently,the nonconvex functions(the smoothly clipped absolute deviation,the minimax con-cave penalty,the capped li-norm penalty and the lp quasi-norm with 0<p<1)have been shown remarkable advantages in variable selection due to the fact that they can overcome the over-penalization.In this paper,we propose and study a novel Noncon-vex Regular Matrix Regression Minimization,which combines the low-rank and sparse regularzations and nonconvex functions perfectly.The Augmented Lagrangian Method(ALM)is proposed to solve the Nonconvex Regular Matrix Regression Minimization Problem.The resulting subproblems either have closed-form solutions or can be solved by fast solvers,which makes the ALM particularly efficient.In theory,we prove that the sequence generated by the ALM converges to a stationary point when the penalty parameter is above a computable threshold.Extensive numerical experiments illustrate that our proposed Nonconvex Regular Matrix Regression Minimization Model outper-forms the existing ones.