Primal-dual Method For Regularized Matrix Regression

Linear regression is a basic problem of supervised learning,which has rich theoretical basis and is widely used in medical treatment,statistics,education and so on.Traditional linear regression models usually consider covariates of vector structure.When the covariates of the problem have a matrix structure,it is necessary to transform the variables of the matrix structure into vectors,and then use the regression model to deal with the vectors.However,direct vectorization will not only cause the problem that the dimension is larger than the sampling,but also destroy the inherent matrix structure information of variables,which makes it difficult to identify the joint effect of travel and column,which brings considerable difficulties to the calculation of regression coefficient.Therefore,it is of great significance to study the linear regression with covariate matrix structure.In this paper,we mainly study the estimation of regression coefficient in linear model with covariate as matrix and response variable as vector.The first work of this paper is to use primal dual method to solve the matrix regression problem of total variation regularization.Firstly,the total variation norm is expressed in dual way,and the minimization problem of total variation matrix regression is transformed into minimax problem.Then the first-order primal dual algorithm is used to solve the minimax problem and estimate the regression coefficient.The second work is to propose a mixed regularized matrix regression model,using the linear combination of the total variation norm and L1 norm as the regularization term,and based on the principle of the first-order primal dual algorithm,a TL1 method is proposed to solve the mixed regularized regression model and estimate the regression coefficient.The experimental results show that the hybrid regularization matrix regression has smaller prediction error than the total variation regularization matrix regression.