A Homotopy Algorithm For Elastic Net

The elastic net model is used as the least squares regularization method.It not only ease of use as the least square method,but also improves the interpretability and prediction accuracy of the model.The elastic net can be widely used in classification problems and regression problems.Since the elastic net is proposed on the basis of ridge regression and Lasso,it inherits the advantages of both.It has good results in the sparse selection of variables and the processing of multicollinearity problems.At the same time,it also overcomes Lasso’s lack of groups effect.Therefore,it is very important to study the elastic net in depth and find a suitable algorithm to solve the elastic net problem.This article mainly starts from the ridge regression and Lasso model definitions and properties,and studies the elastic net model and its solution path.It also analyzes and proves the properties of the elastic net solution path.The research content of this article mainly has the following two aspects:1.Proof The optimality condition of the elastic net solution and the uniqueness of the solution of the elastic net.Based on this,the homotopy algorithm of the elastic net is proposed.At the same time,it is compared and analyzed with the two-stage elastic net algorithm,and numerical experiments are performed using R language programming.The homotopy algorithm of the elastic network has smaller model errors than the two-stage elastic network algorithm.2.Contrast and analyze the performance of the solution path and model error of Lasso,ridge regression and elastic network models under different types of data.Use R language to generate different types of data and use Lasso,ridge regression and elastic network models to solve it,according to the results to compare the errors of their own model,the applicable conditions of the three models are obtained.Chapter 1 of this article briefly introduces the research progress of linear regression,least squares,ridge regression,Lasso and elastic net;Chapter 2 mainly introduces the definition and properties of ridge regression and the definition of Lasso homotopy algorithm of Lasso;Chapter3 Chapter 4 mainly introduces the definition and properties of elastic nets.Chapter 4 proves the optimality conditions and uniqueness of the solutions of elastic nets.The homotopy algorithm of elastic nets is proposed and verified by numerical experiments.Chapter 5 deals with Lasso and ridge regression.A comparative numerical experiment analysis was conducted with the elastic net.Last part are the summary and prospect of the thesis.