{"title":"Sparse Linear Discriminant Analysis Based on lq Regularization","authors":"Chen Jing, Caixia Gao","doi":"10.48014/fcpm.20230529001","DOIUrl":"https://doi.org/10.48014/fcpm.20230529001","url":null,"abstract":"Linear discriminant analysis plays an important role in feature extraction, data dimensionality reduction, and classification. With the progress of science and technology, the data that need to be processed are becoming increasingly large. However, in high-dimensional situations, linear discriminant analysis faces two problems: the lack of interpretability of the projected data since they all involve all p features, which are linear combinations of all features, as well as the singularity problem of the within-class covariance matrix. There are three different arguments for linear discriminant analysis: multivariate Gaussian model, Fisher discrimination problem, and optimal scoring problem. To solve these two problems, this article establishes a model for solving the kth discriminant component, which first transforms the original model of Fisher discriminant problem in linear discriminant analysis by using a diagonal estimated matrix for the within-class variance in place of the original within-class covariance matrix, which overcomes the singularity problem of the matrix and projects it to an orthogonal projection space to remove its orthogonal constraints, and subsequently an lq norm regularization term is added to enhance its interpretability for the purpose of dimensionality reduction and classification. Finally, an iterative algorithm for solving the model and a convergence analysis are given, and it is proved that the sequence generated by the algorithm is descended and converges to a local minimum of the problem for any initial value.","PeriodicalId":343992,"journal":{"name":"Frontiers of Chinese Pure Mathematics","volume":"189 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139334914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Discontinuous Petrov-Galerkin Method for One-Dimensional Hyperbolic Conservation Law Equations Based on HWENO Limiter","authors":"X. Duan, Gao Wei","doi":"10.48014/fcpm.20221019002","DOIUrl":"https://doi.org/10.48014/fcpm.20221019002","url":null,"abstract":"The discontinuous Petrov-Galerkin method (DPG) is a new type of numerical solution to the hyperbolic conservation law equations, which distinguishes itself from the DG method by its high accuracy based on compact stencils. However, in order to overcome the unphysical oscillations of the higher order linear schemes near the large gradient solution, the DPG method often needs to incorporate limiter functions to obtain a high-resolution image of the numerical solution. This paper attempts to combine HWENO as limiter function with DPG to solve the discontinuous initial value problems for the hyperbolic conservation law equations. The single-step high-accuracy SSP Runge-Kutta method is used for time discretization, and a new HWENO-based process is used as the limiter of the RKDPG methods, which requires only one reconstruction to complete the update of the higher-order moments without calculating the linear weight coefficients. . Since the accuracy does not meet the design requirements, the HWENO limiter above is partially improved for the identification of the problem cells in the HWENO limiter above, with the original numerical solution used at the smooth solution. This paper only gives the calculation results of P1 ~P3, and the limiter is also applicable to the DPG method for higher elements. Numerical examples show that the HWENO limiter can effectively suppress non-physical oscillations in the problem cells and keep the original accuracy at the non-problem cells. The high accuracy and compactness characteristics of the DPG method are maintained. The numerical calculation solutions are efficient and accurate.","PeriodicalId":343992,"journal":{"name":"Frontiers of Chinese Pure Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116965520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Nonlinear Scalarization Characterizations of E-Super Efficient Solution of Set-valued Vector Optimization Problem","authors":"Fengya Luo, LI Fei","doi":"10.48014/fcpm.20230420001","DOIUrl":"https://doi.org/10.48014/fcpm.20230420001","url":null,"abstract":"Abstract: In the study of vector optimization problems, the concept of (weakly) effective solution based on the definition of order cone and its properties play a very important role. The concept of superefficient solution is the unification of several existing concept of properly efficient solutions, while the E- super efficient solution of the vector optimization problem proposed based on the improvement set is an important extension of the classical concept of superefficient solution, which unifies the concepts of superefficient solution and ε-super efficient solution. Therefore, it is of great significance to study the E- super efficient solution and its related properties for the study of vector optimization problems. This paper mainly focuses on the relevant conditions of nonlinear scalarization characterization of E- super efficient solution: firstly, a sufficient and necessary condition for the nonlinear scalarization characterization about the E- super efficient solution of the set-valued vector optimization problem is obtained through the corresponding property of the seminorm p; secondly, the corresponding property of the distance function d is used to obtain another sufficient and necessary condition for the nonlinear scalarization characterization of the E- super efficient solution; then, according to the relationship between the E-optimal solution and the E-super efficient solution, a sufficient condition for the results of nonlinear scalarization characterization of the E- super efficient solution is derived by means of the distance-oriented function Δ-E which based on the improved set E under the condition of E=E+K0; finally, under the condition of 0∈clE, the distance-oriented function Δ-E is used to obtain a necessary condition for the nonlinear scalarization result of E- super efficient solution, and the corresponding sufficient and necessary result is deduced.","PeriodicalId":343992,"journal":{"name":"Frontiers of Chinese Pure Mathematics","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114589050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Stationary Distribution of a Random Epidemic Model with Virus Infection","authors":"Zheng Liang","doi":"10.48014/fcpm.20230515001","DOIUrl":"https://doi.org/10.48014/fcpm.20230515001","url":null,"abstract":"Abstract: Since the beginning of 2020 the world has been facing the largest virological invasion in the form. of the COVID-19 pandemic, and the outbreak of COVID-19 has once again demonstrated that infectious diseases remain one of the greatest threats to human survival and development. In this paper, therefore, the existence of a stationary distribution for a class of stochastic COVID-19 infectious disease SEIW (W is the concentration of virus in the environment) models that take into account the effect of environmental viruses is investigated. First, the existence and uniqueness of the solution of the system are proved by constructing a suitable Lyapunov function. The parameters Rs0 are then established using the stochastic Lyapunov method and the existence of a unique stationary distribution of the system solution on R4+ when Rs0 >1 is demonstrated. And by comparing the deterministic model of R0 and the stochastic model of Rs0 , it can be found that Rs0 is influenced by white noise and Rs0 ≤R0, when σi →0 (i=1, 2, 3, 4) , Rs0 →R0, indicating that the work in this paper is an extension of the deterministic model and when the random perturbations are small, there exists a unique stationary distribution on R4+ for the system solution.","PeriodicalId":343992,"journal":{"name":"Frontiers of Chinese Pure Mathematics","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129881700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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