Low-rank And Sparse Regularized Matrix Regression Optimization Method

With the development of science and technology and the advent of the era of big data,a large number of complex data forms have begun to emerge,and there are more and more mixed data containing both matrix and vector forms.Therefore,the matrix regression model of matrix and vector mixed forms gradually has received extensive attention,and it has also been applied to a certain extent in the field of science and practical applications.However,in the process of analyzing these data,we found that some data have a more special structure,such as in video surveillance data,the video data is composed of low-rank and sparse components,that is,low-rank sparse matrix decomposition.In order to emphasize this problem,in this article,we first combine the low-rank sparse matrix decomposition and mixed matrix regression and propose the low-rank sparse mixed matrix regression optimization model.In terms of the nature of the model,we give the weak consistency of the estimated quantities of the model from a statistical perspective,which also guarantees the validity and feasibility of our model from a statistical perspective.In terms of algorithm,we design an effective semiProximal Alternating Direction Method of Multiplier(s PADMM)from the perspective of its duality to solve this dual problem.In the process of solving the iterative steps,the resulting sub-problems either have closed solutions or can be solved by the solution accelerator,which makes the s PADMM algorithm quite effective.In theory,we prove that the iterative sequence generated by the s PADMM algorithm converges to the global optimal solution.Finally,we conducted a large number of numerical experiments from both simulated data and real data.The experiments show that the performance of our proposed low-rank sparse mixed matrix regression optimization method is better than the latest matrix regression minimization method.