Numerical Algorithms On Regression Parameter Estimation Based On Sparsity And Diversity

High-dimensional data or signals often have an important feature:sparsity.Spar-sity means that although the dimension of the sample data is high,most of the infor-mation actually contained is in a lower dimensional subspace.In the era of big data,data sets often have another important feature:diversity.Diversity means the overall datasets do not show a uniform statistical behavior.It may come from a mixture of dif-ferent distributions.The former property emphasizes the characteristics of the sample itself,and the latter analyzes the overall structural characteristics of the samples.For the above two characteristics of data,this dissertation considers three different types of problems:sparse linear regression,1-bit compressed sensing,and regression clustering.For the high-dimensional sparse linear regression problem,we consider the trun-catedl₁regularization model.The non-asymptotic error bound of the global optimal solution to the truncatedl₁problem is obtained.Meanwhile we study the support set recovery properties Based on KKT（Karush-Kuhn-Tucker）conditions to the trun-catedl₁regularization,the primal dual active set algorithm（PDAS）is proposed,and a sequential PDAS version is given.Numerical experiments show that our algorithm is superior to the current popular regularization methods such as LASSO,MCP and SCAD in terms of estimation accuracy,support recovery and computational efficiency.For 1-bit compressed sensing,the cardinality constraint least squares is proposed as a desired decoder.We prove that,up to a constant c,with high probability,the proposed decoder achieves a minimax estimation error as long as m≥O（s log n）.we utilize a generalized Newton algorithm（GNA）to solve the cardinality constraint least square problem.We prove that,with high probability,the estimation error between the output of GNA and the underlying target decays to O(（（log n）/m）^1/2)using at most O（log s）iterations.Extensive numerical simulations and comparisons with state-of-the-art are presented to illustrate the robustness of our proposed decoder and the e ciency of the GNA algorithm.For the problem of regression clustering,based on the idea of fuzzy clustering,combined with thel₁distance and maximum entropy principle,the FCR-l₁-E model is proposed,and the alternating direction multiplier method（ADMM）is used to update the regression coefficients.β_k,the update of the membership degree U_ikis controlled by the entropy penalty coefficient,and meanwhile we give a general rule for the ad-justment of the entropy coefficient.We compare with the simulated data and the real tone data the three representative types of parametric clustering algorithms,and the corresponding version ofl₁models,fully illustrate the robustness of our model against outliers and noise.